Question 1 (Show main steps of your work to get full points)

The \((X'X)^{-1}\) for the \(y=β_0+β_1 x_1+β_2 x_2+β_3 x_3+β_4 x_4+β_5 x_5+β_6 x_6+ε\) is given below.

  1. If MSE = 1.395 and n = 38 , compute the (Keep 4 or more decimal places, DO NOT round in the intermediate steps)

  2. se(β ̂_4)

\[se(\mathbf{\hat\beta_4})=\sqrt{MSE\times C_{55}}=\sqrt{1.395\times0.069}=0.3102499\]

  1. Cov(β ̂_2,β ̂_4)

\[Cov(\mathbf{\hat\beta_2,\hat\beta_4})=MSE\times C_{35}=1.395\times(-0.035)=-0.048825\]

  1. Cor(β ̂_2,β ̂_4 )

\[se(\mathbf{\hat\beta_2})=\sqrt{MSE\times C_{33}}=\sqrt{1.395\times0.067}=0.3057205\]

\[Cor(\mathbf{\hat\beta_2,\hat\beta_4})=\frac{Cov(\mathbf{\hat\beta_2,\hat\beta_4})}{se(\mathbf{\hat\beta_2})se(\mathbf{\hat\beta_4})}=\frac{-0.048825}{0.3057205\times0.3102499}=-0.5147615\]

  1. Without computing anything, explain which estimator is the most consistent.

\(C_{66}=0.058\) has the smallest value. \(\hatβ_5\) has the the least variance and the most consistent among the estimators.

  1. Without computing anything , list the pair(s) of estimators that are positively correlated. Provide a reason.

According to the \((X'X)^{(-1)}\),

\(C_{13},\ C_{17},\ C_{24},\ C_{25},\ C_{67}\) are positive.

The positively correlated pairs of parameters are

\(\hatβ_0\) and \(\hatβ_2\), \(\hatβ_0\) and \(\hatβ_6\), \(\hatβ_1\) and \(\hatβ_3\), \(\hatβ_1\) and \(\hatβ_4\), \(\hatβ_5\) and \(\hatβ_6\).

  1. Consider the following hypothesis: \(H_0: β_1=2β_3,β_2=β_3,β_5=0\)

  2. Report the T matrix, β vector and c vector along with their dimensions, and the rank of T matrix for testing the above hypothesis.

\[ \mathbf{T}=\begin{bmatrix} 0 & 1 & 0 & -2 & 0 & 0& 0 \\ 0 & 0 & 1 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \end{bmatrix}_{3\times7} \mathbf{β}=\begin{bmatrix} \beta_0 \\ \beta_1 \\ \beta_2 \\ \beta_3 \\ \beta_4 \\ \beta_5 \\ \beta_6 \end{bmatrix}_{7\times1} \mathbf{C}=\begin{bmatrix} 0 \\ 0 \\ 0\end{bmatrix}_{3\times1} rank(T)=3 \]

  1. Report the values of numerator and denominator degrees of freedom for the corresponding F test. For F test, the numerator is MSR while denominator is MSE, thus

In this hypothesis,\(y=β_0+2β_3x_1+β_3x_2+β_3x_3+β_4x_4+0x_5+β_6x_6+ε=β_0+β_3(2x_1+x_2+x_3)+β_4x_4+β_6x_6+ε\)

The value of numerator is \(r=df_{Reduced}-df_{Full}=n-(3+1)-[n-(6+1)]=3\)

The denominator degrees of freedom is \(df_{Full}=n-(k+1)=38-(6+1)=31\)

  1. Show the following equation is an alternative form of the sum of squares of regreesion or model (SSR).

\[SSR=\sum_{i=1}^n(\hat y_i-\bar y)^2=\sum_{i=1}^n(\hat y_i^2-2\hat y_i\bar y+\bar y^2)=\sum_{i=1}^n\hat y_i^2-2\bar y\sum_{i=1}^n\hat y_i+\sum_{i=1}^n\bar y^2\]

\[=\sum_{i=1}^n\hat y_i^2-2\bar yn\frac{\sum_{i=1}^n\hat y_i}n+n\bar y^2=\sum_{i=1}^n\hat y_i^2-2\bar yn\bar y+n\bar y^2=\sum_{i=1}^n\hat y_i^2-n\bar y^2\]

Question 2 (Use software to analyze the given data)

The data in the WaterFlow file are simulated data on peak rate of flow (in cfs) of water from six watersheds following storm episodes. The predictors are:

x1 : Area of watershed (mi2) x2 : Area impervious to water (mi2)
x3 : Average slope of watershed (percent)
x4 : Longest stream flow in watershed (1000s of feet)
x5 : surface absorbency index, (0= complete absorbency, 100=no absorbency)
x6 : estimated soil storage capacity (inches of water)
x7 : Infiltration rate of water into soil (inches/hour)
x8 : Rainfall (inches)
x9 : Time period during which rainfall exceeded ¼ inch/hour

  1. Create the matrix of scatterplots and compute the correlation matrix for all the variables. Copy and paste them here.


  1. Based on scatterplots and correlation, explain which predictors are significantly related to (most likely to contribute to the variation in) the response variable.

Based on scatterplots and correlation, X2(0.666),X7(0.668),X1(0.781),X4(0.866) have medium to strong positive linear relationship to the response variable (Correlation coefficient is more than 0.6). X5(-0.62) have medium negative linear relationship to the response variable.


  1. Fit the full model.

\[\hat y=292.561-203.144X_1+ 1055.782X_2-49.24X_3+209.762X_4-10.197X_5-24.558X_6+142.778X_7+511.713X_8-301.872X_9\]


  1. Explain whether the overall model is significant at 5% significance level.

The fitted model is statistically significant at 5% significance level (\(p-value=9.744\times^{-06}\)). But most of the coefficients are not significent. This model is not the best fitted model.


  1. Explain whether assumptions of random errors and model are satisfied. If there is a violation of those, then suggest reasonable methods to correct them.
  • Residual Diagnostics: Use plots to examine residuals to validate OLS assumptions

There is some violation of assumptions about the errors:

On the residual plot, there is a funnel pattern.

On the outlier and leverage plot, there are two outliers.

On the qq plot, most of points follow approximately straight line but have some positive skew.

  • Suggestion: Transform and other diagnostics.

I suggest using natural log of response to make a variance-stabilizing transformations.

Other diagnostics of heteroskedasticity, variable selection, measures of influence also should be considered.


  1. How much of the sum of squares is explained by rainfall, given that all the other regression coefficients are in the model?

Accroding to the F test, the partial sum of squares explained by rainfall is 2209825, given that all the other regression coefficients are in the model.


  1. Explain whether there is a problem of multicollinearity.
  • Collinearity diagnostics:

The model does have serious problems of multicollinearity. The VIF of variables X4 (105.754708), X1 (101.859709), X3 (31.446394), X7(20.53505) are larger than 10.


  1. Interpret the estimated coefficient of rainfall predictor of the full model using question context.

Coefficient of 511.713 suggests the peak rate of flow increases by 511.713 cubic feet per second when the rainfall increases by 1 inch and other variables are constants.


  1. Create a new variable using natural log of response. Then fit the full model using this new variable as response.

\[log(\hat y)=3.402256-0.013532X_1-1.023664X_2+0.177966X_3+0.108788X_4\] \[-0.009622X_5-0.389474X_6+4.233475X_7+0.63007X_8-0.462276X_9\]

  1. Explain whether the overall model is significant at 5% significance level.

The fitted model is statistically significant at 5% significance level (\(p-value=7.513\times10^{-11}\)). But most of the coefficients are not significent. This model is not the best fitted model.


  1. Explain whether there is a problem of multicollinearity.

The model still has serious problems of multicollinearity. The variance-stabilizing transformations does not change the value of VIF.

It will be important to solve multicollinearity. However, X7 (20.53505), X1 (101.859709), and X4 (105.754708) have medium to strong positive linear relationship to the response variable. It is also dangerous to remove these variables. We should have more diagnostics and comparisons.


  1. If you wanted to simplify this full model, explain which predictor you would eliminate first.
  • VIF

If just considering the VIF, X4 (105.754708) or X1 (101.859709) with largest VIF values is the first to remove.

  • Correlation with y

However, according to the correlation coefficients, both X4and X1 is strongly correlated with y (\(Cor_{y,x_4}=0.866,Cor_{y,x_1}=0.781\)) .

The textbook suggest that the general approaches for dealing with multicollinearity include collecting additional data, model respecification (redefine the regressors, variable elimination), estimation methods (Ridge Regression, Principal-Component Regression). “Variable elimination is often a highly effective technique. However, it may not provide a satisfactory solution if the regressors dropped from the model have significant explanatory power relative to the response y. That is, eliminating regressors to reduce multicollinearity may damage the predictive power of the model.” (Montgomery et al., 2012. p.304) In this way, the third multicollinear X3(31.446394) with a weak relationship with y (0.205) should be removed, or even X6(0.0453).

  • Correlation with each other

According to the variable names of X4, X1, and X3, they are geographic variables. Predictor X1 is the area of watershed while X4 is the longest stream flow in watershed, x3 is the average slope of watershed. For the given 6 watersheds, X1 and X4 are strongly related. A high correlation (0.921) is expected between these two variables. But X3 is not significently related with X1(-0.078) or X4(0.245). Removing X3 might lose some irreplacable infromation. I

Actrually, I don’t agree remove any predictor in this stage. Removing any predictor can draw down the VIF significently. After elimination regression, the multicollinearity dissapeared in all the models. We should gather sufficient evidents before removing any predictor.


  1. Use the forward selection method to find the best model (use α=0.15) and report the final fitted model with estimated coefficients here.

Use Stepwise Forward Regression based on p values (use α=0.15)

\[\hat y=2.872+0.168X_3+0.122X_4+3.106X_7\]

Use Stepwise AIC Forwardd Regression

\[\hat y=2.692+0.184X_3+0.109X_4-0.368X_6+4.085X_7+0.612X_8-0.448X_9\]


  1. Use the backward elimination method to find the best model (use α=0.05) and report the final fitted model with estimated coefficients here.

Stepwise Backward Regression based on p values (use α=0.05) and Stepwise AIC Backward Regression have same results.

\[\hat y=2.692+0.184X_3+0.109X_4-0.368X_6+4.085X_7+0.612X_8-0.448X_9\]


  1. Use best subsets method (6 models from each size) to find the best model for these data and report the final fitted model with estimated coefficients here.

Best subsets method gives a same model.

\[\hat y=2.692+0.184X_3+0.109X_4-0.368X_6+4.085X_7+0.612X_8-0.448X_9\]


  1. If the final models in the previous 3 methods are different, compare their model adequacy and suggest one best model.
Method By Keep Remove
Stepwise Forward P=0.15 X3, X4, X7 X1,X2,X5,X6,X8,X9
Stepwise Forward AIC X3,X4,X6,X7,X8,X9 X1,X2,X5
Stepwise Backward P=0.05 X3,X4,X6,X7,X8,X9 X1,X2,X5
Stepwise Backward AIC X3,X4,X6,X7,X8,X9 X1,X2,X5
Stepwise Both P X3, X4, X7 X1,X2,X5,X6,X8,X9
Stepwise Both AIC X3,X4,X6,X7,X8,X9 X1,X2,X5
Best Subset / X3,X4,X6,X7,X8,X9 X1,X2,X5
all possible / X3,X4,X6,X7,X8,X9 X1,X2,X5

Both models solved the problem of multicollinearity (VIF <10), and small P-values for F test. They don’t have serious violation of assumptions about the errors (There is no significant pattern on the plot of studentized residuals versus predicted values from the model with only one predictor. The partial regression plots do not show nonlinear patterns. The points follow approximately straight line on the qq plot). Both of Correlation between observed residuals and expected residuals under normality.The 6-predictor model got 0.9837263 P-value while the 6-predictor model got 0.9856766.

Model VIF F P-value(F) MSR MSE \(R_{adjusted}^2\) \(R_{Predict}^2\) P-value(t) Residuals Plots
3-4-7 <10 70.378 0.0000 21.188 0.301 0.878 0.854 Max=0.054 Good enough
3-4-6-7-8-9 <10 68.16 0.0000 11.265 0.165 0.933 0.908 Max=0.019 Good enough

However, comparing to the 3-variable model, the 6-variable model has a higher (about by 6%) adjusted R square and higher (about by 5%) prediction R-square, which means it shows stronger predictive capability. All the coeficients in 6-predictors model are statistically significant higher than 98% significance level (the maximum p-values are 0.019, respectively). In the 3-variable model, X7 get a high p-value (0.054) which means not significant at 5% significance level. If we change the p-value as the parameter of forward selection, the same model will happened between \(\alpha\) equal 0.6 and 0.17. Further, considering the context, X8 and X9 are variables of precipitation. The 3-predictor model mean the peak flow is irrelevant with precipitation. It doesn’t make sense. Therefore, the best model will be the model with 6 predictors.


  1. Provide complete ANOVA table for the best model. Provide partial sum of squares, estimated coefficients, standard errors, p-values, 95% Bonferroni joint confidence intervals for the coefficients of the best model. Provide in a tabular form clearly.
Model Summary
R 0.973 RMSE (Root Mean Square Error) 0.407
R-Squared 0.947 Coef. Var 6.385
Adj. R-Squared 0.933 MSE (Mean Square Error) 0.165
Pred R-Squared 0.908 MAE (Mean Absolute Error) 0.273
ANOVA
Sum of Squares DF Mean Square F p-value
Regression 67.591 6 11.265 68.16 \(1.717\times10^{-13}\)
Residual 3.801 23 0.165
Total 71.393 29
Parameter Estimates
model Estimated coefficients Partial SS Std. Error t test p-value 2.5 % 97.5 %
(Intercept) 2.69180 / 0.445 6.046 \(3.63\times10{-06}\) 1.77080533 3.61278901
X3 0.18384 5.37 0.032 5.698 \(8.41\times10{-06}\) 0.11709482 0.25059326
X4 0.10905 2.98 0.026 4.244 0.000306 0.05589763 0.16220302
X6 -0.36752 1.05 0.146 -2.526 0.018898 -0.66855123 -0.06648201
X7 4.08497 1.87 1.213 3.367 0.002662 1.57533753 6.59460252
X8 0.61161 3.52 0.133 4.614 0.000122 0.33738118 0.88582991
X9 -0.44764 2.83 0.108 -4.135 0.000402 -0.67159465 -0.22368135

  1. How much variation in the response is explained by the best model after taking number of data and regression coefficients in to account?

By SSR equal 67.591 and SSE equal 3.801, the adjusted R-squared is 0.9329. About 93.29% variation in the response is explained by the best model.


  1. Report the PRESS statistic of the best model.

The value of PRESS is 6.538275. This model explains 90.8% of variation in predicting the peak rate of flow (in cfs) of water from six watersheds following storm episodes.


  1. Report the complete code along with output here.

Singh (1972) used linear models with a logarithm transformation of the variables. We retained the following where the dependent variables can either be total storm flow volume (Qt) in mm, quick flow volume (Qf) in mm or peak flow (Qpk) in m3 sec−1 km−2. Independent variables were storm rainfall (P) in mm, initial flow (Qi) in mm h−1, rainfall frequency (Fp), the inverse of rainfall duration, in h−1 and a dummy variable (CC) representing the treatment effect on basin 7A. CC was 0 and 1 for the calibration (1967–1992) and treated (1994–1998) periods, respectively. β0 to β4 are regression coefficients of the independent variables. All interactions between the independent variables were also tested for significance at α=0.10.

\[ln(Dependent\ Variable)=β_0+β_1lnP+β_2lnQ+β_3lnFP+β_4CC+Interactions+\varepsilon\]

The significance of the regression coefficients (being different from 0) in the models has been tested with a t-test procedure at α=0.10 using the GLM procedure of the SAS system for Windows (SAS Institute, Inc., 1989). A regression coefficient significantly different from 0 for the variable CC indicates that the treatment had a significant effect on the dependent variable. Normality of residuals has been tested using the Shapiro–Wilk test. Selection criteria for all events were the same as for the paired basins approach. However, the rainfall events following night-frost, which may caused localized surface runoff on ice as observed by Prevost et al. (1990), were omitted since these events are too rare to be well represented during the calibration and post-treatment periods. Hence, all events within three weeks following the end of the snowmelt period were not retained. The end of the snowmelt period was obtained from observations in a standard forest snow line at Montmorency Forest. (Guillemette et al., 2005)

  • Residual Diagnostics:

Includes plots to examine residuals to validate OLS assumptions

There is no violation of assumptions about the errors (no pattern on residual plots and points follow approximately straight line on the qq plot).

  • Variable selection:

Differnt variable selection procedures such as all possible regression, best subset regression, stepwise regression, stepwise forward regression and stepwise backward regression

  • Heteroskedasticity:

Tests for heteroskedasticity include bartlett test, breusch pagan test, score test and f test

  • Measures of influence:

Use different plots to detect and identify influential observations

  • Collinearity diagnostics:

VIF, Tolerance and condition indices to detect collinearity and plots for assessing mode fit and contributions of variables

x1 : Area of watershed (mi2)
x8 : Rainfall (inches)

x9 : Time period during which rainfall exceeded ¼ inch/hour
x4 : Longest stream flow in watershed (1000s of feet)

x3 : Average slope of watershed (percent)

x2 : Area impervious to water (mi2)
x5 : surface absorbency index, (0= complete absorbency, 100=no absorbency)
x7 : Infiltration rate of water into soil (inches/hour)
x6 : estimated soil storage capacity (inches of water)

Full model

eliminated model

  • Residual Diagnostics:

Includes plots to examine residuals to validate OLS assumptions

There is no violation of assumptions about the errors (no pattern on residual plots and points follow approximately straight line on the qq plot).

Residual QQ Plot Residual Normality Test Residual vs Fitted Values Plot Residual Histogram

  • Variable selection:

Differnt variable selection procedures such as all possible regression, best subset regression, stepwise regression, stepwise forward regression and stepwise backward regression

  • Heteroskedasticity:

Tests for heteroskedasticity include bartlett test, breusch pagan test, score test and f test

Bartlett Test Breusch Pagan Test Score Test F Test

  • Measures of influence:

Use different plots to detect and identify influential observations

Cook’s D Bar Plot Cook’s D Chart DFBETAs Panel DFFITs Plot Studentized Residual Plot Standardized Residual Chart Studentized Residuals vs Leverage Plot Deleted Studentized Residual vs Fitted Values Plot Hadi Plot Potential Residual Plot

[1]: Montgomery, D. C., Peck, E. A., & Vining, G. G. (2012). Introduction to linear regression analysis (Vol. 821). John Wiley & Sons.

[2]: Guillemette, F., Plamondon, A. P., Prévost, M., & Lévesque, D. (2005). Rainfall generated stormflow response to clearcutting a boreal forest: peak flow comparison with 50 world-wide basin studies. Journal of hydrology, 302(1-4), 137-153.

(a) The matrix of scatterplots and the correlation matrix

library(tidyverse)
table_wf <- read_table2("WaterFlow.txt")
library(GGally)
ggpairs(data=table_wf[c(1:10)])

(c) The fitted full model

# build the model
model_wf_full <- lm(y ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9, data=table_wf)
model_wf_full%>% summary()
## 
## Call:
## lm(formula = y ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9, 
##     data = table_wf)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1404.21  -318.77    74.73   266.66  1274.30 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)  
## (Intercept)   292.56    4428.62   0.066   0.9480  
## X1           -203.14     410.27  -0.495   0.6259  
## X2           1055.78    9833.70   0.107   0.9156  
## X3            -49.24     156.20  -0.315   0.7558  
## X4            209.76     162.05   1.294   0.2103  
## X5            -10.20      51.09  -0.200   0.8438  
## X6            -24.56     303.53  -0.081   0.9363  
## X7            142.78    3288.44   0.043   0.9658  
## X8            511.71     209.74   2.440   0.0241 *
## X9           -301.87     172.00  -1.755   0.0945 .
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 609.3 on 20 degrees of freedom
## Multiple R-squared:  0.8214, Adjusted R-squared:  0.741 
## F-statistic: 10.22 on 9 and 20 DF,  p-value: 9.744e-06
Anova(model_wf_full)
## Anova Table (Type II tests)
## 
## Response: y
##            Sum Sq Df F value  Pr(>F)  
## X1          91022  1  0.2452 0.62589  
## X2           4279  1  0.0115 0.91557  
## X3          36893  1  0.0994 0.75585  
## X4         622091  1  1.6756 0.21025  
## X5          14790  1  0.0398 0.84381  
## X6           2430  1  0.0065 0.93632  
## X7            700  1  0.0019 0.96580  
## X8        2209825  1  5.9523 0.02414 *
## X9        1143622  1  3.0804 0.09455 .
## Residuals 7425127 20                  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(c) ii Residual diagnostics

#Model Fit Assessment
ols_plot_diagnostics(model_wf_full)

# Part & Partial Correlations
ols_test_correlation(model_wf_full) # Correlation between observed residuals and expected residuals under normality.
## [1] 0.9710713
# Residual Normality Test
ols_test_normality(model_wf_full) # Test for detecting violation of normality assumption. #If p-value is bigger, then no problem of non-normality #
## -----------------------------------------------
##        Test             Statistic       pvalue  
## -----------------------------------------------
## Shapiro-Wilk              0.9589         0.2898 
## Kolmogorov-Smirnov        0.1423         0.5314 
## Cramer-von Mises          2.5333         0.0000 
## Anderson-Darling          0.5169         0.1748 
## -----------------------------------------------

(c) iii The partial regression and nonlinear diagnostics

#Lack of Fit F Test
ols_pure_error_anova(lm(y~X8, data = table_wf))
## Lack of Fit F Test 
## ---------------
## Response :   y 
## Predictor:   X8 
## 
##                        Analysis of Variance Table                         
## -------------------------------------------------------------------------
##                 DF      Sum Sq        Mean Sq      F Value       Pr(>F)   
## -------------------------------------------------------------------------
## X8               1     4616882.92    4616882.92    5.795558    0.02290414 
## Residual        28    36951252.44    1319687.59                           
##  Lack of fit    21    31374881.28    1494041.97    1.875466     0.2003839 
##  Pure Error      7     5576371.17     796624.45                           
## -------------------------------------------------------------------------
# Variable Contributions
ols_plot_added_variable(model_wf_full)

# Residual Plus Component Plot
ols_plot_comp_plus_resid(model_wf_full)

(c) iv Collinearity diagnostics

# for full model
ols_vif_tol(model_wf_full)
## # A tibble: 9 x 3
##   Variables Tolerance    VIF
##   <chr>         <dbl>  <dbl>
## 1 X1          0.00982 102.  
## 2 X2          0.133     7.52
## 3 X3          0.0318   31.4 
## 4 X4          0.00946 106.  
## 5 X5          0.103     9.68
## 6 X6          0.433     2.31
## 7 X7          0.0487   20.5 
## 8 X8          0.182     5.50
## 9 X9          0.174     5.75

(d) The fitted log model

# build full log model
model_wf_full_log <- lm(log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9, data=table_wf)
summary(model_wf_full_log)
## 
## Call:
## lm(formula = log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + 
##     X9, data = table_wf)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.95298 -0.20764  0.01499  0.18100  0.67539 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  3.402256   3.150312   1.080 0.293006    
## X1          -0.013532   0.291845  -0.046 0.963477    
## X2          -1.023664   6.995235  -0.146 0.885120    
## X3           0.177966   0.111113   1.602 0.124908    
## X4           0.108788   0.115272   0.944 0.356560    
## X5          -0.009622   0.036341  -0.265 0.793898    
## X6          -0.389474   0.215916  -1.804 0.086345 .  
## X7           4.233475   2.339245   1.810 0.085387 .  
## X8           0.630070   0.149200   4.223 0.000418 ***
## X9          -0.462276   0.122350  -3.778 0.001181 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4334 on 20 degrees of freedom
## Multiple R-squared:  0.9474, Adjusted R-squared:  0.9237 
## F-statistic:    40 on 9 and 20 DF,  p-value: 7.513e-11
Anova(model_wf_full)
## Anova Table (Type II tests)
## 
## Response: y
##            Sum Sq Df F value  Pr(>F)  
## X1          91022  1  0.2452 0.62589  
## X2           4279  1  0.0115 0.91557  
## X3          36893  1  0.0994 0.75585  
## X4         622091  1  1.6756 0.21025  
## X5          14790  1  0.0398 0.84381  
## X6           2430  1  0.0065 0.93632  
## X7            700  1  0.0019 0.96580  
## X8        2209825  1  5.9523 0.02414 *
## X9        1143622  1  3.0804 0.09455 .
## Residuals 7425127 20                  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#Model Fit Assessment
# ols_plot_diagnostics(model_wf_full_log)

# Part & Partial Correlations
# ols_test_correlation(model_wf_full_log) # Correlation between observed residuals and expected residuals under normality.

# Residual Normality Test
# ols_test_normality(model_wf_full_log) # Test for detecting violation of normality assumption. #If p-value is bigger, then no problem of non-normality #
library(dplyr)

(d) (2) Collinearity diagnostics

## Start:  AIC=-42.32
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9
## 
##        Df Sum of Sq    RSS     AIC
## - X1    1    0.0004 3.7577 -44.322
## - X2    1    0.0040 3.7613 -44.293
## - X5    1    0.0132 3.7705 -44.220
## - X4    1    0.1673 3.9246 -43.018
## <none>              3.7573 -42.325
## - X3    1    0.4819 4.2392 -40.705
## - X6    1    0.6113 4.3686 -39.803
## - X7    1    0.6153 4.3726 -39.775
## - X9    1    2.6819 6.4392 -28.164
## - X8    1    3.3503 7.1076 -25.201
## 
## Step:  AIC=-44.32
## log(y) ~ X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9
## 
##        Df Sum of Sq    RSS     AIC
## - X2    1    0.0110 3.7686 -46.234
## - X5    1    0.0267 3.7844 -46.110
## <none>              3.7577 -44.322
## - X6    1    1.0447 4.8023 -38.963
## - X7    1    1.5520 5.3097 -35.950
## - X4    1    1.8469 5.6046 -34.328
## - X9    1    2.8341 6.5918 -29.461
## - X8    1    3.4848 7.2425 -26.637
## - X3    1    5.0955 8.8532 -20.613
## 
## Step:  AIC=-46.23
## log(y) ~ X3 + X4 + X5 + X6 + X7 + X8 + X9
## 
##        Df Sum of Sq    RSS     AIC
## - X5    1    0.0327 3.8013 -47.975
## <none>              3.7686 -46.234
## - X6    1    1.0375 4.8061 -40.939
## - X4    1    1.8741 5.6428 -36.125
## - X7    1    1.9036 5.6722 -35.968
## - X9    1    2.8353 6.6040 -31.406
## - X8    1    3.4744 7.2430 -28.635
## - X3    1    5.1264 8.8951 -22.471
## 
## Step:  AIC=-47.98
## log(y) ~ X3 + X4 + X6 + X7 + X8 + X9
## 
##        Df Sum of Sq    RSS     AIC
## <none>              3.8013 -47.975
## - X6    1    1.0542 4.8555 -42.632
## - X7    1    1.8739 5.6752 -37.953
## - X9    1    2.8256 6.6270 -33.302
## - X4    1    2.9771 6.7784 -32.624
## - X8    1    3.5182 7.3195 -30.320
## - X3    1    5.3653 9.1666 -23.569
## Start:  AIC=-44.32
## log(y) ~ X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9
## 
##        Df Sum of Sq    RSS     AIC
## - X2    1    0.0110 3.7686 -46.234
## - X5    1    0.0267 3.7844 -46.110
## <none>              3.7577 -44.322
## - X6    1    1.0447 4.8023 -38.963
## - X7    1    1.5520 5.3097 -35.950
## - X4    1    1.8469 5.6046 -34.328
## - X9    1    2.8341 6.5918 -29.461
## - X8    1    3.4848 7.2425 -26.637
## - X3    1    5.0955 8.8532 -20.613
## 
## Step:  AIC=-46.23
## log(y) ~ X3 + X4 + X5 + X6 + X7 + X8 + X9
## 
##        Df Sum of Sq    RSS     AIC
## - X5    1    0.0327 3.8013 -47.975
## <none>              3.7686 -46.234
## - X6    1    1.0375 4.8061 -40.939
## - X4    1    1.8741 5.6428 -36.125
## - X7    1    1.9036 5.6722 -35.968
## - X9    1    2.8353 6.6040 -31.406
## - X8    1    3.4744 7.2430 -28.635
## - X3    1    5.1264 8.8951 -22.471
## 
## Step:  AIC=-47.98
## log(y) ~ X3 + X4 + X6 + X7 + X8 + X9
## 
##        Df Sum of Sq    RSS     AIC
## <none>              3.8013 -47.975
## - X6    1    1.0542 4.8555 -42.632
## - X7    1    1.8739 5.6752 -37.953
## - X9    1    2.8256 6.6270 -33.302
## - X4    1    2.9771 6.7784 -32.624
## - X8    1    3.5182 7.3195 -30.320
## - X3    1    5.3653 9.1666 -23.569
## Start:  AIC=-44.29
## log(y) ~ X1 + X3 + X4 + X5 + X6 + X7 + X8 + X9
## 
##        Df Sum of Sq    RSS     AIC
## - X1    1    0.0073 3.7686 -46.234
## - X5    1    0.0370 3.7983 -45.999
## <none>              3.7613 -44.293
## - X4    1    0.3141 4.0754 -43.887
## - X6    1    0.7115 4.4729 -41.095
## - X3    1    0.7775 4.5388 -40.656
## - X7    1    1.2667 5.0280 -37.585
## - X9    1    2.7122 6.4735 -30.005
## - X8    1    3.4001 7.1614 -26.975
## 
## Step:  AIC=-46.23
## log(y) ~ X3 + X4 + X5 + X6 + X7 + X8 + X9
## 
##        Df Sum of Sq    RSS     AIC
## - X5    1    0.0327 3.8013 -47.975
## <none>              3.7686 -46.234
## - X6    1    1.0375 4.8061 -40.939
## - X4    1    1.8741 5.6428 -36.125
## - X7    1    1.9036 5.6722 -35.968
## - X9    1    2.8353 6.6040 -31.406
## - X8    1    3.4744 7.2430 -28.635
## - X3    1    5.1264 8.8951 -22.471
## 
## Step:  AIC=-47.98
## log(y) ~ X3 + X4 + X6 + X7 + X8 + X9
## 
##        Df Sum of Sq    RSS     AIC
## <none>              3.8013 -47.975
## - X6    1    1.0542 4.8555 -42.632
## - X7    1    1.8739 5.6752 -37.953
## - X9    1    2.8256 6.6270 -33.302
## - X4    1    2.9771 6.7784 -32.624
## - X8    1    3.5182 7.3195 -30.320
## - X3    1    5.3653 9.1666 -23.569
## Start:  AIC=-40.7
## log(y) ~ X1 + X2 + X4 + X5 + X6 + X7 + X8 + X9
## 
##        Df Sum of Sq     RSS     AIC
## - X7    1    0.1337  4.3729 -41.773
## - X6    1    0.1858  4.4250 -41.418
## <none>               4.2392 -40.705
## - X2    1    0.2995  4.5388 -40.656
## - X5    1    1.1498  5.3891 -35.505
## - X9    1    3.5480  7.7872 -24.461
## - X8    1    4.0900  8.3292 -22.443
## - X1    1    4.6140  8.8532 -20.613
## - X4    1   13.7481 17.9873   0.654
## 
## Step:  AIC=-41.77
## log(y) ~ X1 + X2 + X4 + X5 + X6 + X8 + X9
## 
##        Df Sum of Sq     RSS     AIC
## - X6    1    0.1369  4.5098 -42.849
## <none>               4.3729 -41.773
## - X2    1    0.6577  5.0306 -39.570
## - X5    1    1.0236  5.3965 -37.464
## - X9    1    3.4161  7.7890 -26.455
## - X8    1    3.9564  8.3293 -24.442
## - X1    1    4.7933  9.1662 -21.570
## - X4    1   13.8200 18.1929  -1.005
## 
## Step:  AIC=-42.85
## log(y) ~ X1 + X2 + X4 + X5 + X8 + X9
## 
##        Df Sum of Sq     RSS     AIC
## <none>               4.5098 -42.849
## - X2    1    0.6110  5.1208 -41.037
## - X5    1    0.8871  5.3969 -39.461
## - X9    1    3.2799  7.7896 -28.452
## - X8    1    3.8347  8.3444 -26.388
## - X1    1    5.0057  9.5155 -22.448
## - X4    1   15.9600 20.4698   0.533
## Start:  AIC=-43.02
## log(y) ~ X1 + X2 + X3 + X5 + X6 + X7 + X8 + X9
## 
##        Df Sum of Sq     RSS     AIC
## - X5    1    0.0457  3.9703 -44.671
## - X2    1    0.1508  4.0754 -43.887
## <none>               3.9246 -43.018
## - X1    1    1.6800  5.6046 -34.328
## - X6    1    2.1914  6.1160 -31.708
## - X9    1    2.5158  6.4404 -30.158
## - X8    1    3.1937  7.1183 -27.156
## - X7    1    4.3217  8.2463 -22.743
## - X3    1   14.0627 17.9873   0.654
## 
## Step:  AIC=-44.67
## log(y) ~ X1 + X2 + X3 + X6 + X7 + X8 + X9
## 
##        Df Sum of Sq     RSS     AIC
## - X2    1    0.1126  4.0829 -45.832
## <none>               3.9703 -44.671
## - X6    1    2.5195  6.4898 -31.929
## - X9    1    2.7581  6.7284 -30.846
## - X1    1    2.7838  6.7541 -30.731
## - X8    1    3.6308  7.6011 -27.187
## - X7    1    4.2769  8.2472 -24.740
## - X3    1   24.3256 28.2959  12.246
## 
## Step:  AIC=-45.83
## log(y) ~ X1 + X3 + X6 + X7 + X8 + X9
## 
##        Df Sum of Sq     RSS     AIC
## <none>               4.0829 -45.832
## - X6    1    2.4147  6.4976 -33.893
## - X9    1    2.6501  6.7330 -32.825
## - X1    1    2.6955  6.7784 -32.624
## - X8    1    3.5347  7.6176 -29.122
## - X7    1    5.2580  9.3409 -23.004
## - X3    1   25.3225 29.4054  11.399
## Start:  AIC=-44.22
## log(y) ~ X1 + X2 + X3 + X4 + X6 + X7 + X8 + X9
## 
##        Df Sum of Sq    RSS     AIC
## - X1    1    0.0139 3.7844 -46.110
## - X2    1    0.0279 3.7983 -45.999
## - X4    1    0.1998 3.9703 -44.671
## <none>              3.7705 -44.220
## - X6    1    0.7524 4.5229 -40.762
## - X7    1    1.1605 4.9309 -38.170
## - X3    1    1.6186 5.3891 -35.505
## - X9    1    2.8181 6.5886 -29.476
## - X8    1    3.5442 7.3147 -26.339
## 
## Step:  AIC=-46.11
## log(y) ~ X2 + X3 + X4 + X6 + X7 + X8 + X9
## 
##        Df Sum of Sq    RSS     AIC
## - X2    1    0.0170 3.8013 -47.975
## <none>              3.7844 -46.110
## - X6    1    1.0707 4.8551 -40.635
## - X7    1    1.5504 5.3348 -37.808
## - X9    1    2.8243 6.6087 -31.384
## - X4    1    2.9697 6.7541 -30.731
## - X8    1    3.5305 7.3149 -28.339
## - X3    1    5.3638 9.1482 -21.629
## 
## Step:  AIC=-47.98
## log(y) ~ X3 + X4 + X6 + X7 + X8 + X9
## 
##        Df Sum of Sq    RSS     AIC
## <none>              3.8013 -47.975
## - X6    1    1.0542 4.8555 -42.632
## - X7    1    1.8739 5.6752 -37.953
## - X9    1    2.8256 6.6270 -33.302
## - X4    1    2.9771 6.7784 -32.624
## - X8    1    3.5182 7.3195 -30.320
## - X3    1    5.3653 9.1666 -23.569
## Start:  AIC=-39.8
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X7 + X8 + X9
## 
##        Df Sum of Sq    RSS     AIC
## - X3    1    0.0564 4.4250 -41.418
## - X2    1    0.1043 4.4729 -41.095
## - X7    1    0.1349 4.5035 -40.891
## - X5    1    0.1543 4.5229 -40.762
## <none>              4.3686 -39.803
## - X1    1    0.4338 4.8023 -38.963
## - X4    1    1.7475 6.1160 -31.708
## - X9    1    2.6668 7.0353 -27.508
## - X8    1    3.1935 7.5620 -25.342
## 
## Step:  AIC=-41.42
## log(y) ~ X1 + X2 + X4 + X5 + X7 + X8 + X9
## 
##        Df Sum of Sq     RSS     AIC
## - X7    1    0.0848  4.5098 -42.849
## - X2    1    0.3043  4.7293 -41.423
## <none>               4.4250 -41.418
## - X5    1    0.9642  5.3892 -37.504
## - X9    1    3.3632  7.7882 -26.458
## - X8    1    3.9187  8.3437 -24.391
## - X1    1    4.6412  9.0662 -21.899
## - X4    1   16.0173 20.4423   2.492
## 
## Step:  AIC=-42.85
## log(y) ~ X1 + X2 + X4 + X5 + X8 + X9
## 
##        Df Sum of Sq     RSS     AIC
## <none>               4.5098 -42.849
## - X2    1    0.6110  5.1208 -41.037
## - X5    1    0.8871  5.3969 -39.461
## - X9    1    3.2799  7.7896 -28.452
## - X8    1    3.8347  8.3444 -26.388
## - X1    1    5.0057  9.5155 -22.448
## - X4    1   15.9600 20.4698   0.533
## Start:  AIC=-39.78
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X8 + X9
## 
##        Df Sum of Sq    RSS     AIC
## - X3    1    0.0003 4.3729 -41.773
## - X6    1    0.1309 4.5035 -40.891
## <none>              4.3726 -39.775
## - X5    1    0.5584 4.9309 -38.170
## - X2    1    0.6554 5.0280 -37.585
## - X1    1    0.9371 5.3097 -35.950
## - X9    1    3.0659 7.4385 -25.836
## - X8    1    3.6837 8.0563 -23.442
## - X4    1    3.8737 8.2463 -22.743
## 
## Step:  AIC=-41.77
## log(y) ~ X1 + X2 + X4 + X5 + X6 + X8 + X9
## 
##        Df Sum of Sq     RSS     AIC
## - X6    1    0.1369  4.5098 -42.849
## <none>               4.3729 -41.773
## - X2    1    0.6577  5.0306 -39.570
## - X5    1    1.0236  5.3965 -37.464
## - X9    1    3.4161  7.7890 -26.455
## - X8    1    3.9564  8.3293 -24.442
## - X1    1    4.7933  9.1662 -21.570
## - X4    1   13.8200 18.1929  -1.005
## 
## Step:  AIC=-42.85
## log(y) ~ X1 + X2 + X4 + X5 + X8 + X9
## 
##        Df Sum of Sq     RSS     AIC
## <none>               4.5098 -42.849
## - X2    1    0.6110  5.1208 -41.037
## - X5    1    0.8871  5.3969 -39.461
## - X9    1    3.2799  7.7896 -28.452
## - X8    1    3.8347  8.3444 -26.388
## - X1    1    5.0057  9.5155 -22.448
## - X4    1   15.9600 20.4698   0.533
## Start:  AIC=-25.2
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X9
## 
##        Df Sum of Sq    RSS     AIC
## - X9    1   0.00002 7.1076 -27.201
## - X4    1   0.01076 7.1183 -27.156
## - X2    1   0.05385 7.1614 -26.975
## - X1    1   0.13496 7.2425 -26.637
## - X5    1   0.20709 7.3147 -26.340
## - X6    1   0.45446 7.5620 -25.342
## <none>              7.1076 -25.201
## - X7    1   0.94872 8.0563 -23.442
## - X3    1   1.22165 8.3292 -22.443
## 
## Step:  AIC=-27.2
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7
## 
##        Df Sum of Sq    RSS     AIC
## - X4    1   0.01127 7.1189 -29.154
## - X2    1   0.05390 7.1615 -28.974
## - X1    1   0.13699 7.2446 -28.628
## - X5    1   0.20769 7.3153 -28.337
## - X6    1   0.45911 7.5667 -27.323
## <none>              7.1076 -27.201
## - X7    1   0.95446 8.0621 -25.421
## - X3    1   1.25421 8.3618 -24.326
## 
## Step:  AIC=-29.15
## log(y) ~ X1 + X2 + X3 + X5 + X6 + X7
## 
##        Df Sum of Sq     RSS      AIC
## - X2    1    0.1409  7.2598 -30.5654
## - X5    1    0.4878  7.6066 -29.1654
## <none>               7.1189 -29.1535
## - X6    1    1.1498  8.2686 -26.6619
## - X1    1    2.6634  9.7823 -21.6187
## - X7    1    3.8818 11.0007 -18.0972
## - X3    1   17.0862 24.2051   5.5609
## 
## Step:  AIC=-30.57
## log(y) ~ X1 + X3 + X5 + X6 + X7
## 
##        Df Sum of Sq     RSS      AIC
## - X5    1    0.3665  7.6263 -31.0878
## <none>               7.2598 -30.5654
## - X6    1    1.1128  8.3726 -28.2870
## - X1    1    2.6550  9.9148 -23.2150
## - X7    1    4.5672 11.8270 -17.9245
## - X3    1   19.3750 26.6348   6.4306
## 
## Step:  AIC=-31.09
## log(y) ~ X1 + X3 + X6 + X7
## 
##        Df Sum of Sq    RSS     AIC
## <none>               7.626 -31.088
## - X6    1    1.5290  9.155 -27.606
## - X1    1    2.7319 10.358 -23.902
## - X7    1    4.8252 12.452 -18.381
## - X3    1   26.0905 33.717  11.504
## Start:  AIC=-28.16
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8
## 
##        Df Sum of Sq    RSS     AIC
## - X4    1   0.00125 6.4404 -30.158
## - X2    1   0.03429 6.4735 -30.005
## - X5    1   0.14939 6.5886 -29.476
## - X1    1   0.15258 6.5918 -29.461
## <none>              6.4392 -28.164
## - X6    1   0.59615 7.0353 -27.508
## - X8    1   0.66842 7.1076 -27.201
## - X7    1   0.99933 7.4385 -25.836
## - X3    1   1.34802 7.7872 -24.462
## 
## Step:  AIC=-30.16
## log(y) ~ X1 + X2 + X3 + X5 + X6 + X7 + X8
## 
##        Df Sum of Sq     RSS     AIC
## - X2    1    0.0670  6.5074 -31.848
## - X5    1    0.2880  6.7284 -30.846
## <none>               6.4404 -30.158
## - X8    1    0.6784  7.1189 -29.153
## - X6    1    1.2983  7.7387 -26.649
## - X1    1    2.0749  8.5153 -23.780
## - X7    1    3.5965 10.0369 -18.848
## - X3    1   16.2255 22.6659   5.590
## 
## Step:  AIC=-31.85
## log(y) ~ X1 + X3 + X5 + X6 + X7 + X8
## 
##        Df Sum of Sq     RSS     AIC
## - X5    1    0.2255  6.7330 -32.825
## <none>               6.5074 -31.848
## - X8    1    0.7523  7.2598 -30.565
## - X6    1    1.2782  7.7856 -28.468
## - X1    1    2.1911  8.6986 -25.141
## - X7    1    4.5538 11.0612 -17.933
## - X3    1   18.9765 25.4839   7.105
## 
## Step:  AIC=-32.83
## log(y) ~ X1 + X3 + X6 + X7 + X8
## 
##        Df Sum of Sq    RSS     AIC
## <none>               6.733 -32.825
## - X8    1    0.8934  7.626 -31.088
## - X6    1    1.6647  8.398 -28.197
## - X1    1    2.4940  9.227 -25.372
## - X7    1    4.7585 11.491 -18.788
## - X3    1   26.5284 33.261  13.096
# Compare vif
ols_vif_tol(model_wf_full_log)
## # A tibble: 9 x 3
##   Variables Tolerance    VIF
##   <chr>         <dbl>  <dbl>
## 1 X1          0.00982 102.  
## 2 X2          0.133     7.52
## 3 X3          0.0318   31.4 
## 4 X4          0.00946 106.  
## 5 X5          0.103     9.68
## 6 X6          0.433     2.31
## 7 X7          0.0487   20.5 
## 8 X8          0.182     5.50
## 9 X9          0.174     5.75
ols_vif_tol(model_wf_aic_log)
## # A tibble: 6 x 3
##   Variables Tolerance   VIF
##   <chr>         <dbl> <dbl>
## 1 X3            0.332  3.01
## 2 X4            0.167  5.97
## 3 X6            0.839  1.19
## 4 X7            0.159  6.28
## 5 X8            0.202  4.94
## 6 X9            0.195  5.12
ols_vif_tol(model_wf_rm1_log)
## # A tibble: 8 x 3
##   Variables Tolerance   VIF
##   <chr>         <dbl> <dbl>
## 1 X2            0.245  4.08
## 2 X3            0.318  3.14
## 3 X4            0.115  8.70
## 4 X5            0.283  3.54
## 5 X6            0.717  1.39
## 6 X7            0.118  8.46
## 7 X8            0.190  5.27
## 8 X9            0.185  5.41
ols_vif_tol(model_wf_rm1_aic_log)
## # A tibble: 6 x 3
##   Variables Tolerance   VIF
##   <chr>         <dbl> <dbl>
## 1 X3            0.332  3.01
## 2 X4            0.167  5.97
## 3 X6            0.839  1.19
## 4 X7            0.159  6.28
## 5 X8            0.202  4.94
## 6 X9            0.195  5.12
ols_vif_tol(model_wf_rm2_log)
## # A tibble: 8 x 3
##   Variables Tolerance   VIF
##   <chr>         <dbl> <dbl>
## 1 X1           0.0181 55.2 
## 2 X3           0.0583 17.2 
## 3 X4           0.0148 67.4 
## 4 X5           0.163   6.13
## 5 X6           0.543   1.84
## 6 X7           0.114   8.79
## 7 X8           0.183   5.46
## 8 X9           0.175   5.72
ols_vif_tol(model_wf_rm2_aic_log)
## # A tibble: 6 x 3
##   Variables Tolerance   VIF
##   <chr>         <dbl> <dbl>
## 1 X3            0.332  3.01
## 2 X4            0.167  5.97
## 3 X6            0.839  1.19
## 4 X7            0.159  6.28
## 5 X8            0.202  4.94
## 6 X9            0.195  5.12
ols_vif_tol(model_wf_rm3_log)
## # A tibble: 8 x 3
##   Variables Tolerance   VIF
##   <chr>         <dbl> <dbl>
## 1 X1           0.0983 10.2 
## 2 X2           0.243   4.11
## 3 X4           0.113   8.87
## 4 X5           0.272   3.68
## 5 X6           0.767   1.30
## 6 X7           0.206   4.85
## 7 X8           0.190   5.26
## 8 X9           0.187   5.36
ols_vif_tol(model_wf_rm3_aic_log)
## # A tibble: 6 x 3
##   Variables Tolerance   VIF
##   <chr>         <dbl> <dbl>
## 1 X1            0.119  8.39
## 2 X2            0.313  3.19
## 3 X4            0.123  8.12
## 4 X5            0.343  2.92
## 5 X8            0.209  4.79
## 6 X9            0.205  4.88
ols_vif_tol(model_wf_rm4_log)
## # A tibble: 8 x 3
##   Variables Tolerance   VIF
##   <chr>         <dbl> <dbl>
## 1 X1            0.119  8.38
## 2 X2            0.209  4.79
## 3 X3            0.379  2.64
## 4 X5            0.187  5.35
## 5 X6            0.836  1.20
## 6 X7            0.165  6.05
## 7 X8            0.187  5.35
## 8 X9            0.183  5.45
ols_vif_tol(model_wf_rm4_aic_log)
## # A tibble: 6 x 3
##   Variables Tolerance   VIF
##   <chr>         <dbl> <dbl>
## 1 X1            0.279  3.58
## 2 X3            0.768  1.30
## 3 X6            0.917  1.09
## 4 X7            0.251  3.99
## 5 X8            0.202  4.94
## 6 X9            0.196  5.10
ols_vif_tol(model_wf_rm5_log)
## # A tibble: 8 x 3
##   Variables Tolerance   VIF
##   <chr>         <dbl> <dbl>
## 1 X1           0.0269 37.2 
## 2 X2           0.210   4.77
## 3 X3           0.0836 12.0 
## 4 X4           0.0171 58.4 
## 5 X6           0.485   2.06
## 6 X7           0.0774 12.9 
## 7 X8           0.200   5.01
## 8 X9           0.190   5.25
ols_vif_tol(model_wf_rm5_aic_log)
## # A tibble: 6 x 3
##   Variables Tolerance   VIF
##   <chr>         <dbl> <dbl>
## 1 X3            0.332  3.01
## 2 X4            0.167  5.97
## 3 X6            0.839  1.19
## 4 X7            0.159  6.28
## 5 X8            0.202  4.94
## 6 X9            0.195  5.12
ols_vif_tol(model_wf_rm6_log)
## # A tibble: 8 x 3
##   Variables Tolerance   VIF
##   <chr>         <dbl> <dbl>
## 1 X1           0.0162 61.5 
## 2 X2           0.167   6.00
## 3 X3           0.0563 17.8 
## 4 X4           0.0182 54.8 
## 5 X5           0.116   8.64
## 6 X7           0.0832 12.0 
## 7 X8           0.182   5.49
## 8 X9           0.174   5.75
ols_vif_tol(model_wf_rm6_aic_log)
## # A tibble: 6 x 3
##   Variables Tolerance   VIF
##   <chr>         <dbl> <dbl>
## 1 X1            0.119  8.39
## 2 X2            0.313  3.19
## 3 X4            0.123  8.12
## 4 X5            0.343  2.92
## 5 X8            0.209  4.79
## 6 X9            0.205  4.88
ols_vif_tol(model_wf_rm7_log)
## # A tibble: 8 x 3
##   Variables Tolerance   VIF
##   <chr>         <dbl> <dbl>
## 1 X1           0.0238 42.0 
## 2 X2           0.310   3.22
## 3 X3           0.135   7.42
## 4 X4           0.0321 31.1 
## 5 X5           0.164   6.09
## 6 X6           0.740   1.35
## 7 X8           0.184   5.45
## 8 X9           0.177   5.66
ols_vif_tol(model_wf_rm7_aic_log)
## # A tibble: 6 x 3
##   Variables Tolerance   VIF
##   <chr>         <dbl> <dbl>
## 1 X1            0.119  8.39
## 2 X2            0.313  3.19
## 3 X4            0.123  8.12
## 4 X5            0.343  2.92
## 5 X8            0.209  4.79
## 6 X9            0.205  4.88
ols_vif_tol(model_wf_rm8_log)
## # A tibble: 8 x 3
##   Variables Tolerance    VIF
##   <chr>         <dbl>  <dbl>
## 1 X1          0.0103   97.5 
## 2 X2          0.134     7.46
## 3 X3          0.0333   30.0 
## 4 X4          0.00973 103.  
## 5 X5          0.114     8.81
## 6 X6          0.435     2.30
## 7 X7          0.0492   20.3 
## 8 X9          0.879     1.14
ols_vif_tol(model_wf_rm8_aic_log)
## # A tibble: 4 x 3
##   Variables Tolerance   VIF
##   <chr>         <dbl> <dbl>
## 1 X1            0.281  3.56
## 2 X3            0.773  1.29
## 3 X6            0.949  1.05
## 4 X7            0.252  3.96
ols_vif_tol(model_wf_rm9_log)
## # A tibble: 8 x 3
##   Variables Tolerance    VIF
##   <chr>         <dbl>  <dbl>
## 1 X1          0.0104   95.8 
## 2 X2          0.134     7.48
## 3 X3          0.0341   29.3 
## 4 X4          0.00998 100.  
## 5 X5          0.113     8.83
## 6 X6          0.433     2.31
## 7 X7          0.0495   20.2 
## 8 X8          0.919     1.09
ols_vif_tol(model_wf_rm9_aic_log)
## # A tibble: 5 x 3
##   Variables Tolerance   VIF
##   <chr>         <dbl> <dbl>
## 1 X1            0.280  3.58
## 2 X3            0.771  1.30
## 3 X6            0.945  1.06
## 4 X7            0.252  3.97
## 5 X8            0.963  1.04

(d) (3) Variable selection

library(huxtable)
huxreg(model_wf_rm1_log, model_wf_rm2_log, model_wf_rm3_log, model_wf_rm4_log, model_wf_rm5_log, model_wf_rm6_log, model_wf_rm7_log, model_wf_rm8_log, model_wf_rm9_log, model_wf_full_log)
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
(Intercept) 3.280     3.703     7.731 *** 1.200     2.581 *** 5.523     7.487 **  -0.182  -0.127   3.402    
(1.690)    (2.333)    (1.678)    (2.110)    (0.547)    (3.075)    (2.314)    (4.072) (3.844)  (3.150)   
X2 -1.243              6.526     -5.001     -2.144     4.655     8.550     -3.730  -2.980   -1.024    
(5.025)             (5.358)    (5.568)    (5.445)    (6.574)    (4.819)    (9.350) (8.912)  (6.995)   
X3 0.183 *** 0.167 *            0.278 *** 0.201 **  0.046     0.002     0.277  0.287 * 0.178    
(0.034)    (0.080)             (0.032)    (0.067)    (0.088)    (0.057)    (0.146) (0.137)  (0.111)   
X4 0.104 **  0.119     0.286 ***          0.088     0.253 **  0.284 *** 0.027  0.009   0.109    
(0.032)    (0.090)    (0.035)             (0.084)    (0.087)    (0.066)    (0.153) (0.143)  (0.115)   
X5 -0.008     -0.013     -0.055 *   0.013              -0.031     -0.050     0.036  0.031   -0.010    
(0.021)    (0.028)    (0.023)    (0.027)             (0.036)    (0.030)    (0.047) (0.044)  (0.036)   
X6 -0.396 *   -0.375     -0.161     -0.531 **  -0.408              -0.138     -0.335  -0.385   -0.389    
(0.164)    (0.188)    (0.168)    (0.155)    (0.199)             (0.174)    (0.289) (0.276)  (0.216)   
X7 4.317 **  3.975 *   0.959     6.088 *** 4.611 *   1.517              5.230  5.352   4.233    
(1.466)    (1.495)    (1.178)    (1.266)    (1.814)    (1.884)             (3.124) (2.965)  (2.339)   
X8 0.629 *** 0.632 *** 0.680 *** 0.606 *** 0.618 *** 0.614 *** 0.657 ***       0.125   0.630 ***
(0.142)    (0.145)    (0.151)    (0.147)    (0.139)    (0.157)    (0.156)          (0.085)  (0.149)   
X9 -0.461 *** -0.464 *** -0.513 *** -0.436 **  -0.453 *** -0.461 **  -0.490 *** 0.001         -0.462 ** 
(0.116)    (0.119)    (0.122)    (0.119)    (0.114)    (0.129)    (0.128)    (0.073)        (0.122)   
X1          -0.042     -0.457 *** 0.250 **  0.048     -0.345     -0.418 *   0.242  0.255   -0.014    
         (0.210)    (0.096)    (0.083)    (0.173)    (0.239)    (0.197)    (0.383) (0.362)  (0.292)   
N 30         30         30         30         30         30         30         30      30       30        
R2 0.947     0.947     0.941     0.945     0.947     0.939     0.939     0.900  0.910   0.947    
logLik -11.407     -11.422     -13.216     -12.059     -11.458     -13.667     -13.681     -20.968  -19.486   -11.406    
AIC 42.815     42.843     46.432     44.118     42.916     47.333     47.361     61.935  58.972   44.811    
*** p < 0.001; ** p < 0.01; * p < 0.05.
huxreg(model_wf_rm1_aic_log, model_wf_rm2_aic_log, model_wf_rm3_aic_log, model_wf_rm4_aic_log, model_wf_rm5_aic_log, model_wf_rm6_aic_log, model_wf_rm7_aic_log, model_wf_rm8_aic_log, model_wf_rm9_aic_log, model_wf_aic_log)
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
(Intercept) 2.692 *** 2.692 *** 6.882 *** 2.307 *** 2.692 *** 6.882 *** 6.882 *** 2.587 *** 2.225 *** 2.692 ***
(0.445)    (0.445)    (1.432)    (0.410)    (0.445)    (1.432)    (1.432)    (0.494)    (0.515)    (0.445)   
X3 0.184 *** 0.184 ***          0.263 *** 0.184 ***                   0.266 *** 0.268 *** 0.184 ***
(0.032)    (0.032)             (0.022)    (0.032)                      (0.029)    (0.028)    (0.032)   
X4 0.109 *** 0.109 *** 0.294 ***          0.109 *** 0.294 *** 0.294 ***                   0.109 ***
(0.026)    (0.026)    (0.033)             (0.026)    (0.033)    (0.033)                      (0.026)   
X6 -0.368 *   -0.368 *            -0.532 **  -0.368 *                     -0.416 *   -0.435 *   -0.368 *  
(0.146)    (0.146)             (0.144)    (0.146)                      (0.186)    (0.179)    (0.146)   
X7 4.085 **  4.085 **           5.453 *** 4.085 **                    5.209 *** 5.174 *** 4.085 ** 
(1.213)    (1.213)             (1.002)    (1.213)                      (1.310)    (1.256)    (1.213)   
X8 0.612 *** 0.612 *** 0.629 *** 0.613 *** 0.612 *** 0.629 *** 0.629 ***          0.141     0.612 ***
(0.133)    (0.133)    (0.142)    (0.137)    (0.133)    (0.142)    (0.142)             (0.079)    (0.133)   
X9 -0.448 *** -0.448 *** -0.471 *** -0.433 *** -0.448 *** -0.471 *** -0.471 ***                   -0.448 ***
(0.108)    (0.108)    (0.115)    (0.112)    (0.108)    (0.115)    (0.115)                      (0.108)   
X1                   -0.432 *** 0.207 ***          -0.432 *** -0.432 *** 0.208 **  0.199 **          
                  (0.086)    (0.053)             (0.086)    (0.086)    (0.070)    (0.067)            
X2                   8.217                       8.217     8.217                               
                  (4.655)                      (4.655)    (4.655)                              
X5                   -0.043 *                     -0.043 *   -0.043 *                             
                  (0.020)                      (0.020)    (0.020)                              
N 30         30         30         30         30         30         30         30         30         30        
R2 0.947     0.947     0.937     0.943     0.947     0.937     0.937     0.893     0.906     0.947    
logLik -11.581     -11.581     -14.144     -12.652     -11.581     -14.144     -14.144     -22.024     -20.155     -11.581    
AIC 39.161     39.161     44.288     41.305     39.161     44.288     44.288     56.049     54.311     39.161    
*** p < 0.001; ** p < 0.01; * p < 0.05.

(d) (4) Forward selection

Stepwise Forward Regression for full model

# Stepwise Forward Regression based on p values (use a=0.15) #
ols_step_forward_p(model_wf_full_log, penter = 0.15)

# Stepwise AIC Forward Regression #
ols_step_forward_aic(model_wf_full_log)

Stepwise Forward Regression for X4 eliminated model

# Stepwise Forward Regression based on p values (use a=0.15) #
ols_step_forward_p(model_wf_rm4_log, penter = 0.15)
# Stepwise AIC Forward Regression #
ols_step_forward_aic(model_wf_rm4_log)

Stepwise Forward Regression for X1 eliminated model

# Stepwise Forward Regression based on p values (use a=0.15) #
ols_step_forward_p(model_wf_rm1_log, penter = 0.15)
# Stepwise AIC Forward Regression #
ols_step_forward_aic(model_wf_rm1_log)

(d) (5) Backward selection

Stepwise Backward Regression for full model

# Stepwise Backward Regression based on p values (use a=0.05) #
ols_step_backward_p(model_wf_full_log, penter = 0.05)
## Backward Elimination Method 
## ---------------------------
## 
## Candidate Terms: 
## 
## 1 . X1 
## 2 . X2 
## 3 . X3 
## 4 . X4 
## 5 . X5 
## 6 . X6 
## 7 . X7 
## 8 . X8 
## 9 . X9 
## 
## We are eliminating variables based on p value...
## 
## Variables Removed: 
## 
## - X1 
## - X2 
## - X5 
## 
## No more variables satisfy the condition of p value = 0.3
## 
## 
## Final Model Output 
## ------------------
## 
##                         Model Summary                         
## -------------------------------------------------------------
## R                       0.973       RMSE               0.407 
## R-Squared               0.947       Coef. Var          6.385 
## Adj. R-Squared          0.933       MSE                0.165 
## Pred R-Squared          0.908       MAE                0.273 
## -------------------------------------------------------------
##  RMSE: Root Mean Square Error 
##  MSE: Mean Square Error 
##  MAE: Mean Absolute Error 
## 
##                                ANOVA                                
## -------------------------------------------------------------------
##                Sum of                                              
##               Squares        DF    Mean Square      F         Sig. 
## -------------------------------------------------------------------
## Regression     67.591         6         11.265     68.16    0.0000 
## Residual        3.801        23          0.165                     
## Total          71.393        29                                    
## -------------------------------------------------------------------
## 
##                                   Parameter Estimates                                    
## ----------------------------------------------------------------------------------------
##       model      Beta    Std. Error    Std. Beta      t        Sig      lower     upper 
## ----------------------------------------------------------------------------------------
## (Intercept)     2.692         0.445                  6.046    0.000     1.771     3.613 
##          X3     0.184         0.032        0.476     5.698    0.000     0.117     0.251 
##          X4     0.109         0.026        0.499     4.244    0.000     0.056     0.162 
##          X6    -0.368         0.146       -0.133    -2.526    0.019    -0.669    -0.066 
##          X7     4.085         1.213        0.406     3.367    0.003     1.575     6.595 
##          X8     0.612         0.133        0.493     4.614    0.000     0.337     0.886 
##          X9    -0.448         0.108       -0.450    -4.135    0.000    -0.672    -0.224 
## ----------------------------------------------------------------------------------------
## 
## 
##                           Elimination Summary                           
## -----------------------------------------------------------------------
##         Variable                  Adj.                                     
## Step    Removed     R-Square    R-Square     C(p)       AIC       RMSE     
## -----------------------------------------------------------------------
##    1    X1            0.9474      0.9273    8.0021    42.8146    0.4230    
##    2    X2            0.9472      0.9304    6.0604    40.9019    0.4139    
##    3    X5            0.9468      0.9329    4.2345    39.1611    0.4065    
## -----------------------------------------------------------------------
# Stepwise AIC Backward Regression #
ols_step_backward_aic(model_wf_full_log)
## Backward Elimination Method 
## ---------------------------
## 
## Candidate Terms: 
## 
## 1 . X1 
## 2 . X2 
## 3 . X3 
## 4 . X4 
## 5 . X5 
## 6 . X6 
## 7 . X7 
## 8 . X8 
## 9 . X9 
## 
## 
## Variables Removed: 
## 
## - X1 
## - X2 
## - X5 
## 
## No more variables to be removed.
## 
## 
##                   Backward Elimination Summary                   
## ---------------------------------------------------------------
## Variable       AIC       RSS     Sum Sq     R-Sq      Adj. R-Sq 
## ---------------------------------------------------------------
## Full Model    44.811    3.757    67.635    0.94737      0.92369 
## X1            42.815    3.758    67.635    0.94737      0.92731 
## X2            40.902    3.769    67.624    0.94721      0.93042 
## X5            39.161    3.801    67.591    0.94675      0.93286 
## ---------------------------------------------------------------

Stepwise Backward Regression for X4 eliminated model

# Stepwise Backward Regression based on p values (use a=0.05) #
ols_step_backward_p(model_wf_rm4_log, penter = 0.05)
# Stepwise AIC Backward Regression #
ols_step_backward_aic(model_wf_rm4_log)

Stepwise Backward Regression for X1 eliminated model

# Stepwise Backward Regression based on p values (use a=0.05) #
ols_step_backward_p(model_wf_rm1_log, penter = 0.05)
# Stepwise AIC Backward Regression #
ols_step_backward_aic(model_wf_rm1_log)

(d) (6) Best Subset Regression

# For full model #
k <- ols_step_best_subset(model_wf_full_log)
k
mindex n predictors rsquare adjr predrsq cp aic sbic sbc msep fpe apc hsp
1 1 X4 0.803 0.796 0.772 48.9  68.4 -19.8 72.6 0.538 0.536 0.225  0.0186 
2 2 X3 X4 0.873 0.864 0.844 24.2  57.2 -30.5 62.8 0.373 0.369 0.155  0.0129 
3 3 X3 X4 X7 0.89  0.878 0.854 19.7  54.8 -32.7 61.8 0.348 0.341 0.143  0.012  
4 4 X1 X4 X8 X9 0.921 0.908 0.886 10.1  47   -38.1 55.4 0.272 0.264 0.111  0.00941
5 5 X3 X4 X7 X8 X9 0.932 0.918 0.892 7.85 44.5 -38.8 54.3 0.255 0.243 0.102  0.0088 
6 6 X3 X4 X6 X7 X8 X9 0.947 0.933 0.908 4.23 39.2 -39.7 50.4 0.217 0.204 0.0857 0.00751
7 7 X3 X4 X5 X6 X7 X8 X9 0.947 0.93  0.902 6.06 40.9 -36.8 53.5 0.236 0.217 0.0912 0.00816
8 8 X2 X3 X4 X5 X6 X7 X8 X9 0.947 0.927 0.896 8    42.8 -33.8 56.8 0.259 0.233 0.0977 0.00895
9 9 X1 X2 X3 X4 X5 X6 X7 X8 X9 0.947 0.924 0.886 10    44.8 -30.8 60.2 0.286 0.25  0.105  0.00989
plot(k)

# For X4 eliminated model #
# k <- ols_step_best_subset(model_wf_rm4_log)
# k
# plot(k)

# For X1 eliminated model #
# k <- ols_step_best_subset(model_wf_rm1_log)
# k
# plot(k)

(d) (7) Models Comparison

  • Model 437896
# build model 437896
model_wf_437896_log <- lm(log(y) ~ X4 + X3 + X7 + X8 + X9 + X6, data=table_wf)
ols_regress(model_wf_437896_log)
##                         Model Summary                         
## -------------------------------------------------------------
## R                       0.973       RMSE               0.407 
## R-Squared               0.947       Coef. Var          6.385 
## Adj. R-Squared          0.933       MSE                0.165 
## Pred R-Squared          0.908       MAE                0.273 
## -------------------------------------------------------------
##  RMSE: Root Mean Square Error 
##  MSE: Mean Square Error 
##  MAE: Mean Absolute Error 
## 
##                                ANOVA                                
## -------------------------------------------------------------------
##                Sum of                                              
##               Squares        DF    Mean Square      F         Sig. 
## -------------------------------------------------------------------
## Regression     67.591         6         11.265     68.16    0.0000 
## Residual        3.801        23          0.165                     
## Total          71.393        29                                    
## -------------------------------------------------------------------
## 
##                                   Parameter Estimates                                    
## ----------------------------------------------------------------------------------------
##       model      Beta    Std. Error    Std. Beta      t        Sig      lower     upper 
## ----------------------------------------------------------------------------------------
## (Intercept)     2.692         0.445                  6.046    0.000     1.771     3.613 
##          X4     0.109         0.026        0.499     4.244    0.000     0.056     0.162 
##          X3     0.184         0.032        0.476     5.698    0.000     0.117     0.251 
##          X7     4.085         1.213        0.406     3.367    0.003     1.575     6.595 
##          X8     0.612         0.133        0.493     4.614    0.000     0.337     0.886 
##          X9    -0.448         0.108       -0.450    -4.135    0.000    -0.672    -0.224 
##          X6    -0.368         0.146       -0.133    -2.526    0.019    -0.669    -0.066 
## ----------------------------------------------------------------------------------------
confint(model_wf_437896_log, level=0.95/1) # Bonferroni joint confidence interval #
##                   2.5 %      97.5 %
## (Intercept)  1.77080533  3.61278901
## X4           0.05589763  0.16220302
## X3           0.11709482  0.25059326
## X7           1.57533753  6.59460252
## X8           0.33738118  0.88582991
## X9          -0.67159465 -0.22368135
## X6          -0.66855123 -0.06648201
# Collinearity Diagnostics #
ols_vif_tol(model_wf_437896_log)
Variables Tolerance VIF
X4 0.167 5.97
X3 0.332 3.01
X7 0.159 6.28
X8 0.202 4.94
X9 0.195 5.12
X6 0.839 1.19
#Model Fit Assessment
ols_plot_diagnostics(model_wf_437896_log)

# Part & Partial Correlations
ols_test_correlation(model_wf_437896_log) # Correlation between observed residuals and expected residuals under normality.
## [1] 0.9837263
# Residual Normality Test
ols_test_normality(model_wf_437896_log) # Test for detecting violation of normality assumption. #If p-value is bigger, then no problem of non-normality #
## -----------------------------------------------
##        Test             Statistic       pvalue  
## -----------------------------------------------
## Shapiro-Wilk              0.9728         0.6175 
## Kolmogorov-Smirnov        0.0997         0.8982 
## Cramer-von Mises          4.8429         0.0000 
## Anderson-Darling          0.2996         0.5612 
## -----------------------------------------------
# Variable Contributions
ols_plot_added_variable(model_wf_437896_log)

# Residual Plus Component Plot
ols_plot_comp_plus_resid(model_wf_437896_log)

  • Model 437
# build model 437
model_wf_437_log <- lm(log(y) ~ X4 + X3 + X7, data=table_wf)
ols_regress(model_wf_437_log)
##                         Model Summary                         
## -------------------------------------------------------------
## R                       0.944       RMSE               0.549 
## R-Squared               0.890       Coef. Var          8.618 
## Adj. R-Squared          0.878       MSE                0.301 
## Pred R-Squared          0.854       MAE                0.414 
## -------------------------------------------------------------
##  RMSE: Root Mean Square Error 
##  MSE: Mean Square Error 
##  MAE: Mean Absolute Error 
## 
##                                ANOVA                                
## -------------------------------------------------------------------
##                Sum of                                              
##               Squares        DF    Mean Square      F         Sig. 
## -------------------------------------------------------------------
## Regression     63.565         3         21.188    70.378    0.0000 
## Residual        7.828        26          0.301                     
## Total          71.393        29                                    
## -------------------------------------------------------------------
## 
##                                  Parameter Estimates                                  
## -------------------------------------------------------------------------------------
##       model     Beta    Std. Error    Std. Beta      t       Sig      lower    upper 
## -------------------------------------------------------------------------------------
## (Intercept)    2.872         0.547                 5.254    0.000     1.748    3.995 
##          X4    0.122         0.033        0.559    3.730    0.001     0.055    0.189 
##          X3    0.168         0.040        0.435    4.165    0.000     0.085    0.251 
##          X7    3.106         1.537        0.309    2.021    0.054    -0.053    6.266 
## -------------------------------------------------------------------------------------
# Collinearity Diagnostics #
ols_vif_tol(model_wf_437_log)
Variables Tolerance VIF
X4 0.188 5.32
X3 0.386 2.59
X7 0.181 5.53
#Model Fit Assessment
ols_plot_diagnostics(model_wf_437_log)

# Part & Partial Correlations
ols_test_correlation(model_wf_437_log) # Correlation between observed residuals and expected residuals under normality.
## [1] 0.9856766
# Residual Normality Test
ols_test_normality(model_wf_437_log) # Test for detecting violation of normality assumption. #If p-value is bigger, then no problem of non-normality #
## -----------------------------------------------
##        Test             Statistic       pvalue  
## -----------------------------------------------
## Shapiro-Wilk              0.9765         0.7267 
## Kolmogorov-Smirnov        0.1033         0.8736 
## Cramer-von Mises          3.1908         0.0000 
## Anderson-Darling          0.3511         0.4469 
## -----------------------------------------------
# Variable Contributions
ols_plot_added_variable(model_wf_437_log)

# Residual Plus Component Plot
ols_plot_comp_plus_resid(model_wf_437_log)

  • Model 137689
# build model 137689
model_wf_137689_log <- lm(log(y) ~ X1 + X3 + X7 + X6 + X8 + X9, data=table_wf)
ols_regress(model_wf_137689_log)
##                         Model Summary                         
## -------------------------------------------------------------
## R                       0.971       RMSE               0.421 
## R-Squared               0.943       Coef. Var          6.618 
## Adj. R-Squared          0.928       MSE                0.178 
## Pred R-Squared          0.900       MAE                0.292 
## -------------------------------------------------------------
##  RMSE: Root Mean Square Error 
##  MSE: Mean Square Error 
##  MAE: Mean Absolute Error 
## 
##                                ANOVA                                
## -------------------------------------------------------------------
##                Sum of                                              
##               Squares        DF    Mean Square      F         Sig. 
## -------------------------------------------------------------------
## Regression     67.310         6         11.218    63.195    0.0000 
## Residual        4.083        23          0.178                     
## Total          71.393        29                                    
## -------------------------------------------------------------------
## 
##                                   Parameter Estimates                                    
## ----------------------------------------------------------------------------------------
##       model      Beta    Std. Error    Std. Beta      t        Sig      lower     upper 
## ----------------------------------------------------------------------------------------
## (Intercept)     2.307         0.410                  5.623    0.000     1.458     3.156 
##          X1     0.207         0.053        0.368     3.897    0.001     0.097     0.317 
##          X3     0.263         0.022        0.680    11.944    0.000     0.217     0.308 
##          X7     5.453         1.002        0.542     5.442    0.000     3.380     7.525 
##          X6    -0.532         0.144       -0.192    -3.688    0.001    -0.831    -0.234 
##          X8     0.613         0.137        0.495     4.462    0.000     0.329     0.897 
##          X9    -0.433         0.112       -0.435    -3.864    0.001    -0.665    -0.201 
## ----------------------------------------------------------------------------------------
# Collinearity Diagnostics #
ols_vif_tol(model_wf_137689_log)
Variables Tolerance VIF
X1 0.279 3.58
X3 0.768 1.3 
X7 0.251 3.99
X6 0.917 1.09
X8 0.202 4.94
X9 0.196 5.1 
#Model Fit Assessment
ols_plot_diagnostics(model_wf_137689_log)

# Part & Partial Correlations
ols_test_correlation(model_wf_137689_log) # Correlation between observed residuals and expected residuals under normality.
## [1] 0.988106
# Residual Normality Test
ols_test_normality(model_wf_137689_log) # Test for detecting violation of normality assumption. #If p-value is bigger, then no problem of non-normality #
## -----------------------------------------------
##        Test             Statistic       pvalue  
## -----------------------------------------------
## Shapiro-Wilk              0.9769         0.7382 
## Kolmogorov-Smirnov        0.0771         0.9881 
## Cramer-von Mises          4.4689         0.0000 
## Anderson-Darling          0.1644         0.9350 
## -----------------------------------------------
# Variable Contributions
ols_plot_added_variable(model_wf_137689_log)

# Residual Plus Component Plot
ols_plot_comp_plus_resid(model_wf_137689_log)

# Check PRESS Statistic
ols_press(model_wf_full)
## [1] 15880486
ols_press(model_wf_full_log)
## [1] 8.136733
ols_press(model_wf_437896_log)
## [1] 6.538275
ols_press(model_wf_437_log)
## [1] 10.43262
ols_press(model_wf_137689_log)
## [1] 7.114336
# prediction power
ols_pred_rsq(model_wf_full)
## [1] 0.6179649
ols_pred_rsq(model_wf_full_log)
## [1] 0.8860283
ols_pred_rsq(model_wf_437896_log)
## [1] 0.908418
ols_pred_rsq(model_wf_437_log)
## [1] 0.8538697
ols_pred_rsq(model_wf_137689_log)
## [1] 0.900349

More

  • Other Models
# build X1*X8 eliminated log model
model_wf_18rm4_log <- lm(log(y) ~ X1*X8 + X3 + X6 + X7 + X9, data=table_wf)

# build X1*X8 eliminated log model
table_wf_resi <- table_wf%>% mutate(x1t8=X1*X8)
model_wf_1time8_log <- lm(log(y) ~ x1t8 + X3 + X6 + X7+ X9 , data=table_wf_resi)

# build X1*X4 eliminated log model
table_wf_resi <- table_wf%>% mutate(x1t4=X1*X4)
model_wf_1time4_log <- lm(log(y) ~ x1t4 + X3 + X6 + X7+ X8+ X9, data=table_wf_resi)
summary(model_wf_1time4_log)

# build X1/X4 eliminated log model
table_wf_resi <- table_wf%>% mutate(x14=X1/X4)
model_wf_1per4_log <- lm(log(y) ~ x14 + X3 + X6 + X7+ X8+ X9, data=table_wf_resi)

# build X4*X3 eliminated log model
model_wf_43rm1_log <- lm(log(y) ~ X9 + X4*X3 + X6 + X7 + X8 , data=table_wf)

# build X4*X9 eliminated log model
model_wf_49rm1_log <- lm(log(y) ~ X3 + X4*X9 + X6 + X7 + X8 , data=table_wf)

# build X4*X9 eliminated log model
model_wf_48rm1_log <- lm(log(y) ~ X3 + X4*X8 + X6 + X7 + X9 , data=table_wf)

# build X4*X9 eliminated log model
model_wf_47rm1_log <- lm(log(y) ~ X3 + X4*X7 + X6 + X9 + X8 , data=table_wf)

# build X4/X9 eliminated log model
table_wf_resi <- table_wf%>% mutate(x4p9=X4/X9)
model_wf_4per9_log <- lm(log(y) ~ X3 + x4p9 + X6 + X7 + X8 , data=table_wf_resi)

# build X3/X4vX8*X9 eliminated log model
model_wf_34v89_log <- lm(log(y) ~ X3*X4 + X8*X9 + X6 + X7, data=table_wf_resi)

# build X3/X4vX8*X9 eliminated log model
model_wf_34v89v67_log <- lm(log(y) ~ X3*X4 + X8*X9 + X6*X7, data=table_wf_resi)

# build X8/X9vX4*X3 eliminated log model
table_wf_resi <- table_wf%>% mutate(x8p9=X8/X9)
model_wf_8per9v43_log <- lm(log(y) ~ x8p9 + X4*X3 + X6 + X7, data=table_wf_resi)

# build X6/7vX8/X9vX4X3 eliminated log model
table_wf_resi <- table_wf%>% mutate(x8p9=X8/X9,x6p7=X6/X7)
model_wf_6p7v8p9v43_log <- lm(log(y) ~ x8p9 + X4*X3 + x6p7, data=table_wf_resi)

# build X8/X9vX4*X3rmX7 eliminated log model
table_wf_resi <- table_wf%>% mutate(x8p9=X8/X9)
model_wf_8per9v43rm7_log <- lm(log(y) ~ x8p9 + X4*X3 + X6, data=table_wf_resi)

# build X8/X9vX4*X3vX6/X7 eliminated log model
table_wf_resi <- table_wf%>% mutate(x8p9=X8/X9)
model_wf_8per9v43rm7_log <- lm(log(y) ~ x8p9 + X4*X3 + X6, data=table_wf_resi)

huxreg(model_wf_8per9v43rm7_log, model_wf_8per9v43_log, model_wf_43rm1_log, model_wf_6p7v8p9v43_log, model_wf_34v89_log, model_wf_34v89v67_log)
  • Interaction models
# Interaction regression for full
model_wf_full_log_inter <- lm(log(y)~ (X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9)^2, data=table_wf)
model_wf_aic_log_inter <- stepAIC(model_wf_full_log_inter)
## Start:  AIC=-86.68
## log(y) ~ (X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9)^2
## 
## 
## Step:  AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 + 
##     X1:X3 + X1:X4 + X1:X5 + X1:X6 + X1:X7 + X1:X8 + X1:X9 + X2:X3 + 
##     X2:X4 + X2:X5 + X2:X6 + X2:X7 + X2:X8 + X2:X9 + X3:X4 + X3:X5 + 
##     X3:X6 + X3:X7 + X3:X8 + X3:X9 + X4:X5 + X4:X6 + X4:X7 + X4:X8 + 
##     X4:X9 + X5:X6 + X5:X7 + X5:X8 + X5:X9 + X6:X8 + X6:X9 + X7:X8 + 
##     X7:X9 + X8:X9
## 
## 
## Step:  AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 + 
##     X1:X3 + X1:X4 + X1:X5 + X1:X6 + X1:X7 + X1:X8 + X1:X9 + X2:X3 + 
##     X2:X4 + X2:X5 + X2:X6 + X2:X7 + X2:X8 + X2:X9 + X3:X4 + X3:X5 + 
##     X3:X6 + X3:X7 + X3:X8 + X3:X9 + X4:X5 + X4:X6 + X4:X7 + X4:X8 + 
##     X4:X9 + X5:X6 + X5:X8 + X5:X9 + X6:X8 + X6:X9 + X7:X8 + X7:X9 + 
##     X8:X9
## 
## 
## Step:  AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 + 
##     X1:X3 + X1:X4 + X1:X5 + X1:X6 + X1:X7 + X1:X8 + X1:X9 + X2:X3 + 
##     X2:X4 + X2:X5 + X2:X6 + X2:X7 + X2:X8 + X2:X9 + X3:X4 + X3:X5 + 
##     X3:X6 + X3:X7 + X3:X8 + X3:X9 + X4:X5 + X4:X6 + X4:X7 + X4:X8 + 
##     X4:X9 + X5:X8 + X5:X9 + X6:X8 + X6:X9 + X7:X8 + X7:X9 + X8:X9
## 
## 
## Step:  AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 + 
##     X1:X3 + X1:X4 + X1:X5 + X1:X6 + X1:X7 + X1:X8 + X1:X9 + X2:X3 + 
##     X2:X4 + X2:X5 + X2:X6 + X2:X7 + X2:X8 + X2:X9 + X3:X4 + X3:X5 + 
##     X3:X6 + X3:X7 + X3:X8 + X3:X9 + X4:X5 + X4:X6 + X4:X8 + X4:X9 + 
##     X5:X8 + X5:X9 + X6:X8 + X6:X9 + X7:X8 + X7:X9 + X8:X9
## 
## 
## Step:  AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 + 
##     X1:X3 + X1:X4 + X1:X5 + X1:X6 + X1:X7 + X1:X8 + X1:X9 + X2:X3 + 
##     X2:X4 + X2:X5 + X2:X6 + X2:X7 + X2:X8 + X2:X9 + X3:X4 + X3:X5 + 
##     X3:X6 + X3:X7 + X3:X8 + X3:X9 + X4:X5 + X4:X8 + X4:X9 + X5:X8 + 
##     X5:X9 + X6:X8 + X6:X9 + X7:X8 + X7:X9 + X8:X9
## 
## 
## Step:  AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 + 
##     X1:X3 + X1:X4 + X1:X5 + X1:X6 + X1:X7 + X1:X8 + X1:X9 + X2:X3 + 
##     X2:X4 + X2:X5 + X2:X6 + X2:X7 + X2:X8 + X2:X9 + X3:X4 + X3:X5 + 
##     X3:X6 + X3:X7 + X3:X8 + X3:X9 + X4:X8 + X4:X9 + X5:X8 + X5:X9 + 
##     X6:X8 + X6:X9 + X7:X8 + X7:X9 + X8:X9
## 
## 
## Step:  AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 + 
##     X1:X3 + X1:X4 + X1:X5 + X1:X6 + X1:X7 + X1:X8 + X1:X9 + X2:X3 + 
##     X2:X4 + X2:X5 + X2:X6 + X2:X7 + X2:X8 + X2:X9 + X3:X4 + X3:X5 + 
##     X3:X6 + X3:X8 + X3:X9 + X4:X8 + X4:X9 + X5:X8 + X5:X9 + X6:X8 + 
##     X6:X9 + X7:X8 + X7:X9 + X8:X9
## 
## 
## Step:  AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 + 
##     X1:X3 + X1:X4 + X1:X5 + X1:X6 + X1:X7 + X1:X8 + X1:X9 + X2:X3 + 
##     X2:X4 + X2:X5 + X2:X6 + X2:X7 + X2:X8 + X2:X9 + X3:X4 + X3:X5 + 
##     X3:X8 + X3:X9 + X4:X8 + X4:X9 + X5:X8 + X5:X9 + X6:X8 + X6:X9 + 
##     X7:X8 + X7:X9 + X8:X9
## 
## 
## Step:  AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 + 
##     X1:X3 + X1:X4 + X1:X5 + X1:X6 + X1:X7 + X1:X8 + X1:X9 + X2:X3 + 
##     X2:X4 + X2:X5 + X2:X6 + X2:X7 + X2:X8 + X2:X9 + X3:X4 + X3:X8 + 
##     X3:X9 + X4:X8 + X4:X9 + X5:X8 + X5:X9 + X6:X8 + X6:X9 + X7:X8 + 
##     X7:X9 + X8:X9
## 
## 
## Step:  AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 + 
##     X1:X3 + X1:X4 + X1:X5 + X1:X6 + X1:X7 + X1:X8 + X1:X9 + X2:X3 + 
##     X2:X4 + X2:X5 + X2:X6 + X2:X7 + X2:X8 + X2:X9 + X3:X8 + X3:X9 + 
##     X4:X8 + X4:X9 + X5:X8 + X5:X9 + X6:X8 + X6:X9 + X7:X8 + X7:X9 + 
##     X8:X9
## 
## 
## Step:  AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 + 
##     X1:X3 + X1:X4 + X1:X5 + X1:X6 + X1:X7 + X1:X8 + X1:X9 + X2:X3 + 
##     X2:X4 + X2:X5 + X2:X6 + X2:X8 + X2:X9 + X3:X8 + X3:X9 + X4:X8 + 
##     X4:X9 + X5:X8 + X5:X9 + X6:X8 + X6:X9 + X7:X8 + X7:X9 + X8:X9
## 
## 
## Step:  AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 + 
##     X1:X3 + X1:X4 + X1:X5 + X1:X6 + X1:X7 + X1:X8 + X1:X9 + X2:X3 + 
##     X2:X4 + X2:X5 + X2:X8 + X2:X9 + X3:X8 + X3:X9 + X4:X8 + X4:X9 + 
##     X5:X8 + X5:X9 + X6:X8 + X6:X9 + X7:X8 + X7:X9 + X8:X9
## 
## 
## Step:  AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 + 
##     X1:X3 + X1:X4 + X1:X5 + X1:X6 + X1:X7 + X1:X8 + X1:X9 + X2:X3 + 
##     X2:X4 + X2:X8 + X2:X9 + X3:X8 + X3:X9 + X4:X8 + X4:X9 + X5:X8 + 
##     X5:X9 + X6:X8 + X6:X9 + X7:X8 + X7:X9 + X8:X9
## 
## 
## Step:  AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 + 
##     X1:X3 + X1:X4 + X1:X5 + X1:X6 + X1:X7 + X1:X8 + X1:X9 + X2:X3 + 
##     X2:X8 + X2:X9 + X3:X8 + X3:X9 + X4:X8 + X4:X9 + X5:X8 + X5:X9 + 
##     X6:X8 + X6:X9 + X7:X8 + X7:X9 + X8:X9
## 
## 
## Step:  AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 + 
##     X1:X3 + X1:X4 + X1:X5 + X1:X6 + X1:X7 + X1:X8 + X1:X9 + X2:X8 + 
##     X2:X9 + X3:X8 + X3:X9 + X4:X8 + X4:X9 + X5:X8 + X5:X9 + X6:X8 + 
##     X6:X9 + X7:X8 + X7:X9 + X8:X9
## 
## 
## Step:  AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 + 
##     X1:X3 + X1:X4 + X1:X5 + X1:X6 + X1:X8 + X1:X9 + X2:X8 + X2:X9 + 
##     X3:X8 + X3:X9 + X4:X8 + X4:X9 + X5:X8 + X5:X9 + X6:X8 + X6:X9 + 
##     X7:X8 + X7:X9 + X8:X9
## 
## 
## Step:  AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 + 
##     X1:X3 + X1:X4 + X1:X5 + X1:X8 + X1:X9 + X2:X8 + X2:X9 + X3:X8 + 
##     X3:X9 + X4:X8 + X4:X9 + X5:X8 + X5:X9 + X6:X8 + X6:X9 + X7:X8 + 
##     X7:X9 + X8:X9
## 
## 
## Step:  AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 + 
##     X1:X3 + X1:X4 + X1:X8 + X1:X9 + X2:X8 + X2:X9 + X3:X8 + X3:X9 + 
##     X4:X8 + X4:X9 + X5:X8 + X5:X9 + X6:X8 + X6:X9 + X7:X8 + X7:X9 + 
##     X8:X9
## 
## 
## Step:  AIC=-86.68
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 + 
##     X1:X3 + X1:X8 + X1:X9 + X2:X8 + X2:X9 + X3:X8 + X3:X9 + X4:X8 + 
##     X4:X9 + X5:X8 + X5:X9 + X6:X8 + X6:X9 + X7:X8 + X7:X9 + X8:X9
## 
##         Df Sum of Sq     RSS     AIC
## - X1:X8  1   0.00125 0.27702 -88.546
## - X3:X9  1   0.00166 0.27743 -88.501
## - X1:X9  1   0.00171 0.27748 -88.496
## - X4:X9  1   0.00224 0.27801 -88.439
## - X5:X8  1   0.00375 0.27952 -88.276
## - X3:X8  1   0.01365 0.28942 -87.232
## - X5:X9  1   0.01394 0.28971 -87.202
## <none>               0.27577 -86.682
## - X4:X8  1   0.01926 0.29503 -86.656
## - X8:X9  1   0.02380 0.29957 -86.198
## - X1:X2  1   0.02492 0.30069 -86.086
## - X2:X8  1   0.02521 0.30098 -86.057
## - X6:X8  1   0.02975 0.30552 -85.608
## - X6:X9  1   0.03024 0.30601 -85.560
## - X2:X9  1   0.03404 0.30981 -85.190
## - X7:X8  1   0.04050 0.31627 -84.570
## - X7:X9  1   0.08581 0.36158 -80.554
## - X1:X3  1   1.65959 1.93536 -30.227
## 
## Step:  AIC=-88.55
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 + 
##     X1:X3 + X1:X9 + X2:X8 + X2:X9 + X3:X8 + X3:X9 + X4:X8 + X4:X9 + 
##     X5:X8 + X5:X9 + X6:X8 + X6:X9 + X7:X8 + X7:X9 + X8:X9
## 
##         Df Sum of Sq     RSS     AIC
## - X4:X9  1   0.00103 0.27806 -90.434
## - X5:X8  1   0.00630 0.28332 -89.871
## - X3:X9  1   0.01467 0.29170 -88.997
## <none>               0.27702 -88.546
## - X5:X9  1   0.01990 0.29692 -88.465
## - X2:X8  1   0.02658 0.30360 -87.797
## - X1:X2  1   0.02706 0.30408 -87.750
## - X3:X8  1   0.02955 0.30658 -87.504
## - X6:X9  1   0.03164 0.30866 -87.301
## - X6:X8  1   0.03412 0.31114 -87.061
## - X4:X8  1   0.03459 0.31162 -87.015
## - X2:X9  1   0.03623 0.31325 -86.859
## - X8:X9  1   0.03768 0.31470 -86.720
## - X7:X8  1   0.04267 0.31969 -86.248
## - X1:X9  1   0.08036 0.35738 -82.904
## - X7:X9  1   0.08949 0.36652 -82.147
## - X1:X3  1   1.67908 1.95611 -31.907
## 
## Step:  AIC=-90.43
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 + 
##     X1:X3 + X1:X9 + X2:X8 + X2:X9 + X3:X8 + X3:X9 + X4:X8 + X5:X8 + 
##     X5:X9 + X6:X8 + X6:X9 + X7:X8 + X7:X9 + X8:X9
## 
##         Df Sum of Sq     RSS     AIC
## - X5:X8  1   0.01021 0.28826 -91.352
## <none>               0.27806 -90.434
## - X3:X9  1   0.01962 0.29768 -90.388
## - X1:X2  1   0.02775 0.30580 -89.580
## - X3:X8  1   0.02855 0.30661 -89.502
## - X5:X9  1   0.02886 0.30692 -89.471
## - X6:X9  1   0.03330 0.31136 -89.040
## - X8:X9  1   0.03752 0.31557 -88.637
## - X6:X8  1   0.04082 0.31888 -88.324
## - X7:X8  1   0.05026 0.32832 -87.449
## - X2:X8  1   0.07559 0.35365 -85.220
## - X1:X9  1   0.08639 0.36445 -84.317
## - X2:X9  1   0.09477 0.37282 -83.635
## - X7:X9  1   0.09547 0.37353 -83.579
## - X4:X8  1   0.11185 0.38991 -82.291
## - X1:X3  1   1.76157 2.03963 -32.653
## 
## Step:  AIC=-91.35
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 + 
##     X1:X3 + X1:X9 + X2:X8 + X2:X9 + X3:X8 + X3:X9 + X4:X8 + X5:X9 + 
##     X6:X8 + X6:X9 + X7:X8 + X7:X9 + X8:X9
## 
##         Df Sum of Sq     RSS     AIC
## - X3:X9  1   0.01387 0.30213 -91.943
## <none>               0.28826 -91.352
## - X1:X2  1   0.02606 0.31432 -90.756
## - X6:X9  1   0.02892 0.31718 -90.484
## - X5:X9  1   0.03517 0.32343 -89.899
## - X6:X8  1   0.03526 0.32352 -89.891
## - X8:X9  1   0.03647 0.32474 -89.778
## - X7:X8  1   0.05418 0.34244 -88.186
## - X3:X8  1   0.06678 0.35505 -87.101
## - X1:X9  1   0.08233 0.37059 -85.815
## - X2:X8  1   0.09026 0.37852 -85.180
## - X4:X8  1   0.11594 0.40420 -83.211
## - X2:X9  1   0.12196 0.41023 -82.767
## - X7:X9  1   0.19579 0.48405 -77.803
## - X1:X3  1   1.79585 2.08412 -34.006
## 
## Step:  AIC=-91.94
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 + 
##     X1:X3 + X1:X9 + X2:X8 + X2:X9 + X3:X8 + X4:X8 + X5:X9 + X6:X8 + 
##     X6:X9 + X7:X8 + X7:X9 + X8:X9
## 
##         Df Sum of Sq     RSS     AIC
## - X6:X9  1   0.01568 0.31781 -92.425
## <none>               0.30213 -91.943
## - X6:X8  1   0.02200 0.32413 -91.834
## - X5:X9  1   0.02250 0.32464 -91.788
## - X1:X2  1   0.03453 0.33666 -90.696
## - X7:X8  1   0.04139 0.34352 -90.092
## - X8:X9  1   0.05081 0.35295 -89.279
## - X1:X9  1   0.07027 0.37241 -87.669
## - X2:X8  1   0.07640 0.37853 -87.180
## - X3:X8  1   0.09504 0.39717 -85.737
## - X4:X8  1   0.10557 0.40770 -84.952
## - X2:X9  1   0.10898 0.41111 -84.703
## - X7:X9  1   0.20828 0.51041 -78.212
## - X1:X3  1   1.80109 2.10322 -35.732
## 
## Step:  AIC=-92.43
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 + 
##     X1:X3 + X1:X9 + X2:X8 + X2:X9 + X3:X8 + X4:X8 + X5:X9 + X6:X8 + 
##     X7:X8 + X7:X9 + X8:X9
## 
##         Df Sum of Sq     RSS     AIC
## - X6:X8  1   0.00642 0.32423 -93.825
## - X1:X2  1   0.01993 0.33773 -92.601
## <none>               0.31781 -92.425
## - X5:X9  1   0.02449 0.34230 -92.198
## - X7:X8  1   0.02573 0.34354 -92.090
## - X8:X9  1   0.03582 0.35363 -91.221
## - X1:X9  1   0.06065 0.37846 -89.186
## - X2:X8  1   0.06122 0.37903 -89.140
## - X3:X8  1   0.08481 0.40262 -87.329
## - X4:X8  1   0.09252 0.41033 -86.760
## - X2:X9  1   0.09419 0.41200 -86.638
## - X7:X9  1   0.23406 0.55187 -77.869
## - X1:X3  1   1.89418 2.21199 -36.219
## 
## Step:  AIC=-93.83
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 + 
##     X1:X3 + X1:X9 + X2:X8 + X2:X9 + X3:X8 + X4:X8 + X5:X9 + X7:X8 + 
##     X7:X9 + X8:X9
## 
##         Df Sum of Sq     RSS     AIC
## - X7:X8  1   0.02050 0.34473 -93.986
## - X1:X2  1   0.02195 0.34617 -93.860
## <none>               0.32423 -93.825
## - X6     1   0.02936 0.35359 -93.225
## - X5:X9  1   0.03666 0.36089 -92.611
## - X8:X9  1   0.05990 0.38412 -90.740
## - X2:X8  1   0.10202 0.42624 -87.618
## - X2:X9  1   0.16870 0.49293 -83.258
## - X7:X9  1   0.23395 0.55817 -79.529
## - X1:X9  1   0.25322 0.57745 -78.510
## - X3:X8  1   0.26381 0.58803 -77.965
## - X4:X8  1   0.40165 0.72587 -71.647
## - X1:X3  1   1.90127 2.22550 -38.036
## 
## Step:  AIC=-93.99
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X2 + 
##     X1:X3 + X1:X9 + X2:X8 + X2:X9 + X3:X8 + X4:X8 + X5:X9 + X7:X9 + 
##     X8:X9
## 
##         Df Sum of Sq     RSS     AIC
## - X1:X2  1   0.01744 0.36216 -94.506
## <none>               0.34473 -93.986
## - X6     1   0.02380 0.36853 -93.983
## - X8:X9  1   0.04553 0.39026 -92.264
## - X5:X9  1   0.04913 0.39386 -91.988
## - X2:X8  1   0.08418 0.42891 -89.432
## - X2:X9  1   0.15080 0.49553 -85.100
## - X1:X9  1   0.27457 0.61930 -78.411
## - X3:X8  1   0.30355 0.64828 -77.039
## - X7:X9  1   0.32337 0.66809 -76.136
## - X4:X8  1   0.42623 0.77096 -71.840
## - X1:X3  1   1.88785 2.23258 -39.941
## 
## Step:  AIC=-94.51
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X1:X3 + 
##     X1:X9 + X2:X8 + X2:X9 + X3:X8 + X4:X8 + X5:X9 + X7:X9 + X8:X9
## 
##         Df Sum of Sq     RSS     AIC
## - X6     1   0.02085 0.38302 -94.826
## <none>               0.36216 -94.506
## - X8:X9  1   0.03703 0.39920 -93.585
## - X5:X9  1   0.03720 0.39936 -93.573
## - X2:X8  1   0.06882 0.43098 -91.286
## - X2:X9  1   0.13369 0.49585 -87.080
## - X1:X9  1   0.26000 0.62217 -80.272
## - X3:X8  1   0.29045 0.65261 -78.839
## - X7:X9  1   0.30630 0.66847 -78.119
## - X4:X8  1   0.40893 0.77109 -73.834
## - X1:X3  1   1.87267 2.23483 -41.911
## 
## Step:  AIC=-94.83
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X7 + X8 + X9 + X1:X3 + X1:X9 + 
##     X2:X8 + X2:X9 + X3:X8 + X4:X8 + X5:X9 + X7:X9 + X8:X9
## 
##         Df Sum of Sq     RSS     AIC
## - X5:X9  1   0.02376 0.40678 -95.021
## - X8:X9  1   0.02579 0.40880 -94.871
## <none>               0.38302 -94.826
## - X2:X8  1   0.04986 0.43287 -93.155
## - X2:X9  1   0.11290 0.49591 -89.077
## - X1:X9  1   0.23967 0.62268 -82.247
## - X3:X8  1   0.28141 0.66443 -80.301
## - X7:X9  1   0.28680 0.66981 -80.059
## - X4:X8  1   0.39670 0.77972 -75.501
## - X1:X3  1   2.28557 2.66859 -38.589
## 
## Step:  AIC=-95.02
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X7 + X8 + X9 + X1:X3 + X1:X9 + 
##     X2:X8 + X2:X9 + X3:X8 + X4:X8 + X7:X9 + X8:X9
## 
##         Df Sum of Sq     RSS     AIC
## - X8:X9  1   0.02190 0.42867 -95.448
## <none>               0.40678 -95.021
## - X2:X8  1   0.02957 0.43635 -94.916
## - X2:X9  1   0.08976 0.49654 -91.039
## - X5     1   0.18602 0.59280 -85.723
## - X7:X9  1   0.26382 0.67060 -82.024
## - X1:X9  1   0.32901 0.73579 -79.240
## - X3:X8  1   0.35450 0.76128 -78.219
## - X4:X8  1   0.43472 0.84149 -75.213
## - X1:X3  1   2.29575 2.70253 -40.210
## 
## Step:  AIC=-95.45
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X7 + X8 + X9 + X1:X3 + X1:X9 + 
##     X2:X8 + X2:X9 + X3:X8 + X4:X8 + X7:X9
## 
##         Df Sum of Sq     RSS     AIC
## - X2:X8  1   0.01037 0.43904 -96.731
## <none>               0.42867 -95.448
## - X2:X9  1   0.06896 0.49763 -92.973
## - X5     1   0.17518 0.60385 -87.169
## - X7:X9  1   0.24199 0.67066 -84.021
## - X1:X9  1   0.30722 0.73589 -81.236
## - X3:X8  1   0.34794 0.77661 -79.620
## - X4:X8  1   0.47278 0.90146 -75.148
## - X1:X3  1   2.41556 2.84423 -40.677
## 
## Step:  AIC=-96.73
## log(y) ~ X1 + X2 + X3 + X4 + X5 + X7 + X8 + X9 + X1:X3 + X1:X9 + 
##     X2:X9 + X3:X8 + X4:X8 + X7:X9
## 
##         Df Sum of Sq     RSS     AIC
## <none>               0.43904 -96.731
## - X5     1   0.17280 0.61185 -88.774
## - X7:X9  1   0.24527 0.68431 -85.416
## - X2:X9  1   0.30033 0.73937 -83.095
## - X1:X9  1   0.31184 0.75088 -82.631
## - X3:X8  1   0.40524 0.84428 -79.114
## - X4:X8  1   0.66804 1.10708 -70.984
## - X1:X3  1   2.49992 2.93897 -41.694
# Interaction regression for remove X1-9
model_wf_rm1_log_inter <- lm(log(y) ~ (X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9)^2, data=table_wf)
model_wf_rm1_aic_log_inter <- stepAIC(model_wf_rm1_log_inter)
model_wf_rm2_log_inter <- lm(log(y) ~ (X1 + X3 + X4 + X5 + X6 + X7 + X8 + X9)^2, data=table_wf)
model_wf_rm2_aic_log_inter <- stepAIC(model_wf_rm2_log_inter)
model_wf_rm3_log_inter <- lm(log(y) ~ (X1 + X2 + X4 + X5 + X6 + X7+ X8 + X9)^2, data=table_wf)
model_wf_rm3_aic_log_inter <- stepAIC(model_wf_rm3_log_inter)
model_wf_rm5_log_inter <- lm(log(y) ~ (X1 + X2 + X3 + X4 + X6 + X7 + X8 + X9)^2, data=table_wf)
model_wf_rm5_aic_log_inter <- stepAIC(model_wf_rm5_log_inter)
model_wf_rm4_log_inter <- lm(log(y) ~ (X1 + X2 + X3 + X5 + X6 + X7 + X8 + X9)^2, data=table_wf)
model_wf_rm4_aic_log_inter <- stepAIC(model_wf_rm4_log_inter)
model_wf_rm6_log_inter <- lm(log(y) ~ (X2 + X3 + X1 + X5 + X4 + X7 + X8 + X9)^2, data=table_wf)
model_wf_rm6_aic_log_inter <- stepAIC(model_wf_rm6_log_inter)
model_wf_rm7_log_inter <- lm(log(y) ~ (X1 + X2 + X3 + X4 + X5 + X6 + X8 + X9)^2, data=table_wf)
model_wf_rm7_aic_log_inter <- stepAIC(model_wf_rm7_log_inter)
model_wf_rm8_log_inter <- lm(log(y) ~ (X1 + X2 + X3 + X4 + X5 + X6 + X7 + X9)^2, data=table_wf)
model_wf_rm8_aic_log_inter <- stepAIC(model_wf_rm8_log_inter)
model_wf_rm9_log_inter <- lm(log(y) ~ (X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8)^2, data=table_wf)
model_wf_rm9_aic_log_inter <- stepAIC(model_wf_rm9_log_inter)

# Interaction regression for 136789
model_wf_136789_log_inter <- lm(log(y) ~ (X3 + X1 + X6 + X7 + X8 + X9)^2, data=table_wf)
model_wf_136789_aic_log_inter <- stepAIC(model_wf_136789_log_inter)
# Interaction regression for 436789
model_wf_436789_log_inter <- lm(log(y) ~ (X3 + X4 + X6 + X7 + X8 + X9)^2, data=table_wf)
model_wf_436789_aic_log_inter <- stepAIC(model_wf_436789_log_inter)
# Interaction regression for 437
model_wf_437_log_inter <- lm(log(y) ~ (X3 + X4 + X7 )^2, data=table_wf)
model_wf_437_aic_log_inter <- stepAIC(model_wf_437_log_inter)

# Interaction regression for 489
model_wf_489_log_inter <- lm(log(y) ~ (X4 + X8 + X9 )^2, data=table_wf)
model_wf_489_aic_log_inter <- stepAIC(model_wf_489_log_inter)

# Interaction regression by groups
model_wf_3g_log_inter <- lm(log(y) ~ (X1 + X3 + X4 )^2 + (X2+ X5+X6 + X7)^2 + (X8 + X9)^2, data=table_wf)
model_wf_3g_aic_log_inter <- stepAIC(model_wf_3g_log_inter)

# Interaction regression by groups1
model_wf_3g1_log_inter <- lm(log(y) ~ (X1 + X2 + X5 )^2 + (X3 +X6 + X7)^2 + (X4 +X8 + X9)^2, data=table_wf)
model_wf_3g1_aic_log_inter <- stepAIC(model_wf_3g1_log_inter)

# Comparison
huxreg(model_wf_rm1_aic_log_inter, model_wf_rm2_aic_log_inter, model_wf_rm3_aic_log_inter, model_wf_rm4_aic_log_inter, model_wf_rm5_aic_log_inter, model_wf_rm6_aic_log_inter, model_wf_rm7_aic_log_inter, model_wf_rm8_aic_log_inter, model_wf_rm9_aic_log_inter, model_wf_aic_log_inter)

huxreg(model_wf_136789_aic_log_inter,model_wf_436789_aic_log_inter, model_wf_437_log_inter, model_wf_489_log_inter, model_wf_3g_aic_log_inter, model_wf_3g1_aic_log_inter)
  • All log models
# build all log model
model_wf_all_log <- lm(log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) + log(X7) + log(X8) + log(X9), data=table_wf)
ols_vif_tol(model_wf_all_log)
Variables Tolerance VIF
log(X1) 0.00608 164   
log(X2) 0.0865  11.6 
log(X3) 0.0816  12.3 
log(X4) 0.00767 130   
log(X5) 0.0885  11.3 
log(X6) 0.421   2.37
log(X7) 0.108   9.29
log(X8) 0.193   5.18
log(X9) 0.187   5.35
model_wf_aic_all_log <- stepAIC(model_wf_all_log)
## Start:  AIC=-84.46
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) + 
##     log(X7) + log(X8) + log(X9)
## 
##           Df Sum of Sq    RSS     AIC
## - log(X7)  1    0.0027 0.9251 -86.371
## - log(X2)  1    0.0310 0.9534 -85.468
## - log(X4)  1    0.0370 0.9595 -85.277
## - log(X3)  1    0.0525 0.9750 -84.797
## - log(X5)  1    0.0574 0.9798 -84.647
## <none>                 0.9224 -84.459
## - log(X6)  1    0.2329 1.1553 -79.705
## - log(X1)  1    0.2712 1.1936 -78.727
## - log(X9)  1    3.4818 4.4043 -39.559
## - log(X8)  1    3.6487 4.5711 -38.443
## 
## Step:  AIC=-86.37
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) + 
##     log(X8) + log(X9)
## 
##           Df Sum of Sq    RSS     AIC
## - log(X4)  1    0.0370 0.9621 -87.193
## - log(X2)  1    0.0559 0.9810 -86.611
## <none>                 0.9251 -86.371
## - log(X3)  1    0.0777 1.0028 -85.953
## - log(X5)  1    0.0983 1.0234 -85.341
## - log(X6)  1    0.3174 1.2425 -79.522
## - log(X1)  1    0.3899 1.3150 -77.820
## - log(X9)  1    3.4793 4.4044 -41.558
## - log(X8)  1    3.6745 4.5996 -40.257
## 
## Step:  AIC=-87.19
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X5) + log(X6) + log(X8) + 
##     log(X9)
## 
##           Df Sum of Sq    RSS     AIC
## - log(X2)  1    0.0369 0.9990 -88.065
## <none>                 0.9621 -87.193
## - log(X5)  1    0.1419 1.1040 -85.067
## - log(X6)  1    0.2985 1.2607 -81.087
## - log(X3)  1    0.5087 1.4709 -76.460
## - log(X9)  1    3.4515 4.4137 -43.495
## - log(X8)  1    3.6420 4.6042 -42.227
## - log(X1)  1    3.8134 4.7755 -41.131
## 
## Step:  AIC=-88.07
## log(y) ~ log(X1) + log(X3) + log(X5) + log(X6) + log(X8) + log(X9)
## 
##           Df Sum of Sq     RSS     AIC
## <none>                  0.9990 -88.065
## - log(X5)  1    0.1087  1.1077 -86.967
## - log(X6)  1    0.3805  1.3795 -80.384
## - log(X3)  1    0.8252  1.8242 -72.001
## - log(X9)  1    3.4549  4.4539 -45.222
## - log(X8)  1    3.7305  4.7295 -43.421
## - log(X1)  1   17.5601 18.5592  -2.407
ols_vif_tol(model_wf_aic_all_log)
Variables Tolerance VIF
log(X1) 0.263 3.8 
log(X3) 0.603 1.66
log(X5) 0.22  4.55
log(X6) 0.71  1.41
log(X8) 0.201 4.99
log(X9) 0.191 5.22
# Interaction regression for all log  model
model_wf_all_log_inter <- lm(log(y) ~ (log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) + log(X7) + log(X8) + log(X9))^2, data=table_wf)
model_wf_aic_all_log_inter <- stepAIC(model_wf_all_log_inter)
## Start:  AIC=-117.75
## log(y) ~ (log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) + 
##     log(X7) + log(X8) + log(X9))^2
## 
## 
## Step:  AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) + 
##     log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) + 
##     log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X6) + log(X1):log(X7) + 
##     log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X3) + log(X2):log(X4) + 
##     log(X2):log(X5) + log(X2):log(X6) + log(X2):log(X7) + log(X2):log(X8) + 
##     log(X2):log(X9) + log(X3):log(X4) + log(X3):log(X5) + log(X3):log(X6) + 
##     log(X3):log(X7) + log(X3):log(X8) + log(X3):log(X9) + log(X4):log(X5) + 
##     log(X4):log(X6) + log(X4):log(X7) + log(X4):log(X8) + log(X4):log(X9) + 
##     log(X5):log(X6) + log(X5):log(X7) + log(X5):log(X8) + log(X5):log(X9) + 
##     log(X6):log(X8) + log(X6):log(X9) + log(X7):log(X8) + log(X7):log(X9) + 
##     log(X8):log(X9)
## 
## 
## Step:  AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) + 
##     log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) + 
##     log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X6) + log(X1):log(X7) + 
##     log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X3) + log(X2):log(X4) + 
##     log(X2):log(X5) + log(X2):log(X6) + log(X2):log(X7) + log(X2):log(X8) + 
##     log(X2):log(X9) + log(X3):log(X4) + log(X3):log(X5) + log(X3):log(X6) + 
##     log(X3):log(X7) + log(X3):log(X8) + log(X3):log(X9) + log(X4):log(X5) + 
##     log(X4):log(X6) + log(X4):log(X7) + log(X4):log(X8) + log(X4):log(X9) + 
##     log(X5):log(X6) + log(X5):log(X8) + log(X5):log(X9) + log(X6):log(X8) + 
##     log(X6):log(X9) + log(X7):log(X8) + log(X7):log(X9) + log(X8):log(X9)
## 
## 
## Step:  AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) + 
##     log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) + 
##     log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X6) + log(X1):log(X7) + 
##     log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X3) + log(X2):log(X4) + 
##     log(X2):log(X5) + log(X2):log(X6) + log(X2):log(X7) + log(X2):log(X8) + 
##     log(X2):log(X9) + log(X3):log(X4) + log(X3):log(X5) + log(X3):log(X6) + 
##     log(X3):log(X7) + log(X3):log(X8) + log(X3):log(X9) + log(X4):log(X5) + 
##     log(X4):log(X6) + log(X4):log(X7) + log(X4):log(X8) + log(X4):log(X9) + 
##     log(X5):log(X8) + log(X5):log(X9) + log(X6):log(X8) + log(X6):log(X9) + 
##     log(X7):log(X8) + log(X7):log(X9) + log(X8):log(X9)
## 
## 
## Step:  AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) + 
##     log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) + 
##     log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X6) + log(X1):log(X7) + 
##     log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X3) + log(X2):log(X4) + 
##     log(X2):log(X5) + log(X2):log(X6) + log(X2):log(X7) + log(X2):log(X8) + 
##     log(X2):log(X9) + log(X3):log(X4) + log(X3):log(X5) + log(X3):log(X6) + 
##     log(X3):log(X7) + log(X3):log(X8) + log(X3):log(X9) + log(X4):log(X5) + 
##     log(X4):log(X6) + log(X4):log(X8) + log(X4):log(X9) + log(X5):log(X8) + 
##     log(X5):log(X9) + log(X6):log(X8) + log(X6):log(X9) + log(X7):log(X8) + 
##     log(X7):log(X9) + log(X8):log(X9)
## 
## 
## Step:  AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) + 
##     log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) + 
##     log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X6) + log(X1):log(X7) + 
##     log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X3) + log(X2):log(X4) + 
##     log(X2):log(X5) + log(X2):log(X6) + log(X2):log(X7) + log(X2):log(X8) + 
##     log(X2):log(X9) + log(X3):log(X4) + log(X3):log(X5) + log(X3):log(X6) + 
##     log(X3):log(X7) + log(X3):log(X8) + log(X3):log(X9) + log(X4):log(X5) + 
##     log(X4):log(X8) + log(X4):log(X9) + log(X5):log(X8) + log(X5):log(X9) + 
##     log(X6):log(X8) + log(X6):log(X9) + log(X7):log(X8) + log(X7):log(X9) + 
##     log(X8):log(X9)
## 
## 
## Step:  AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) + 
##     log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) + 
##     log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X6) + log(X1):log(X7) + 
##     log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X3) + log(X2):log(X4) + 
##     log(X2):log(X5) + log(X2):log(X6) + log(X2):log(X7) + log(X2):log(X8) + 
##     log(X2):log(X9) + log(X3):log(X4) + log(X3):log(X5) + log(X3):log(X6) + 
##     log(X3):log(X7) + log(X3):log(X8) + log(X3):log(X9) + log(X4):log(X8) + 
##     log(X4):log(X9) + log(X5):log(X8) + log(X5):log(X9) + log(X6):log(X8) + 
##     log(X6):log(X9) + log(X7):log(X8) + log(X7):log(X9) + log(X8):log(X9)
## 
## 
## Step:  AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) + 
##     log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) + 
##     log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X6) + log(X1):log(X7) + 
##     log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X3) + log(X2):log(X4) + 
##     log(X2):log(X5) + log(X2):log(X6) + log(X2):log(X7) + log(X2):log(X8) + 
##     log(X2):log(X9) + log(X3):log(X4) + log(X3):log(X5) + log(X3):log(X6) + 
##     log(X3):log(X8) + log(X3):log(X9) + log(X4):log(X8) + log(X4):log(X9) + 
##     log(X5):log(X8) + log(X5):log(X9) + log(X6):log(X8) + log(X6):log(X9) + 
##     log(X7):log(X8) + log(X7):log(X9) + log(X8):log(X9)
## 
## 
## Step:  AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) + 
##     log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) + 
##     log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X6) + log(X1):log(X7) + 
##     log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X3) + log(X2):log(X4) + 
##     log(X2):log(X5) + log(X2):log(X6) + log(X2):log(X7) + log(X2):log(X8) + 
##     log(X2):log(X9) + log(X3):log(X4) + log(X3):log(X5) + log(X3):log(X8) + 
##     log(X3):log(X9) + log(X4):log(X8) + log(X4):log(X9) + log(X5):log(X8) + 
##     log(X5):log(X9) + log(X6):log(X8) + log(X6):log(X9) + log(X7):log(X8) + 
##     log(X7):log(X9) + log(X8):log(X9)
## 
## 
## Step:  AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) + 
##     log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) + 
##     log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X6) + log(X1):log(X7) + 
##     log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X3) + log(X2):log(X4) + 
##     log(X2):log(X5) + log(X2):log(X6) + log(X2):log(X7) + log(X2):log(X8) + 
##     log(X2):log(X9) + log(X3):log(X4) + log(X3):log(X8) + log(X3):log(X9) + 
##     log(X4):log(X8) + log(X4):log(X9) + log(X5):log(X8) + log(X5):log(X9) + 
##     log(X6):log(X8) + log(X6):log(X9) + log(X7):log(X8) + log(X7):log(X9) + 
##     log(X8):log(X9)
## 
## 
## Step:  AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) + 
##     log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) + 
##     log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X6) + log(X1):log(X7) + 
##     log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X3) + log(X2):log(X4) + 
##     log(X2):log(X5) + log(X2):log(X6) + log(X2):log(X7) + log(X2):log(X8) + 
##     log(X2):log(X9) + log(X3):log(X8) + log(X3):log(X9) + log(X4):log(X8) + 
##     log(X4):log(X9) + log(X5):log(X8) + log(X5):log(X9) + log(X6):log(X8) + 
##     log(X6):log(X9) + log(X7):log(X8) + log(X7):log(X9) + log(X8):log(X9)
## 
## 
## Step:  AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) + 
##     log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) + 
##     log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X6) + log(X1):log(X7) + 
##     log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X3) + log(X2):log(X4) + 
##     log(X2):log(X5) + log(X2):log(X6) + log(X2):log(X8) + log(X2):log(X9) + 
##     log(X3):log(X8) + log(X3):log(X9) + log(X4):log(X8) + log(X4):log(X9) + 
##     log(X5):log(X8) + log(X5):log(X9) + log(X6):log(X8) + log(X6):log(X9) + 
##     log(X7):log(X8) + log(X7):log(X9) + log(X8):log(X9)
## 
## 
## Step:  AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) + 
##     log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) + 
##     log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X6) + log(X1):log(X7) + 
##     log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X3) + log(X2):log(X4) + 
##     log(X2):log(X5) + log(X2):log(X8) + log(X2):log(X9) + log(X3):log(X8) + 
##     log(X3):log(X9) + log(X4):log(X8) + log(X4):log(X9) + log(X5):log(X8) + 
##     log(X5):log(X9) + log(X6):log(X8) + log(X6):log(X9) + log(X7):log(X8) + 
##     log(X7):log(X9) + log(X8):log(X9)
## 
## 
## Step:  AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) + 
##     log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) + 
##     log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X6) + log(X1):log(X7) + 
##     log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X3) + log(X2):log(X4) + 
##     log(X2):log(X8) + log(X2):log(X9) + log(X3):log(X8) + log(X3):log(X9) + 
##     log(X4):log(X8) + log(X4):log(X9) + log(X5):log(X8) + log(X5):log(X9) + 
##     log(X6):log(X8) + log(X6):log(X9) + log(X7):log(X8) + log(X7):log(X9) + 
##     log(X8):log(X9)
## 
## 
## Step:  AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) + 
##     log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) + 
##     log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X6) + log(X1):log(X7) + 
##     log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X3) + log(X2):log(X8) + 
##     log(X2):log(X9) + log(X3):log(X8) + log(X3):log(X9) + log(X4):log(X8) + 
##     log(X4):log(X9) + log(X5):log(X8) + log(X5):log(X9) + log(X6):log(X8) + 
##     log(X6):log(X9) + log(X7):log(X8) + log(X7):log(X9) + log(X8):log(X9)
## 
## 
## Step:  AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) + 
##     log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) + 
##     log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X6) + log(X1):log(X7) + 
##     log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X8) + log(X2):log(X9) + 
##     log(X3):log(X8) + log(X3):log(X9) + log(X4):log(X8) + log(X4):log(X9) + 
##     log(X5):log(X8) + log(X5):log(X9) + log(X6):log(X8) + log(X6):log(X9) + 
##     log(X7):log(X8) + log(X7):log(X9) + log(X8):log(X9)
## 
## 
## Step:  AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) + 
##     log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) + 
##     log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X6) + log(X1):log(X8) + 
##     log(X1):log(X9) + log(X2):log(X8) + log(X2):log(X9) + log(X3):log(X8) + 
##     log(X3):log(X9) + log(X4):log(X8) + log(X4):log(X9) + log(X5):log(X8) + 
##     log(X5):log(X9) + log(X6):log(X8) + log(X6):log(X9) + log(X7):log(X8) + 
##     log(X7):log(X9) + log(X8):log(X9)
## 
## 
## Step:  AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) + 
##     log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) + 
##     log(X1):log(X4) + log(X1):log(X5) + log(X1):log(X8) + log(X1):log(X9) + 
##     log(X2):log(X8) + log(X2):log(X9) + log(X3):log(X8) + log(X3):log(X9) + 
##     log(X4):log(X8) + log(X4):log(X9) + log(X5):log(X8) + log(X5):log(X9) + 
##     log(X6):log(X8) + log(X6):log(X9) + log(X7):log(X8) + log(X7):log(X9) + 
##     log(X8):log(X9)
## 
## 
## Step:  AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) + 
##     log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) + 
##     log(X1):log(X4) + log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X8) + 
##     log(X2):log(X9) + log(X3):log(X8) + log(X3):log(X9) + log(X4):log(X8) + 
##     log(X4):log(X9) + log(X5):log(X8) + log(X5):log(X9) + log(X6):log(X8) + 
##     log(X6):log(X9) + log(X7):log(X8) + log(X7):log(X9) + log(X8):log(X9)
## 
## 
## Step:  AIC=-117.75
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) + 
##     log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) + 
##     log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X8) + log(X2):log(X9) + 
##     log(X3):log(X8) + log(X3):log(X9) + log(X4):log(X8) + log(X4):log(X9) + 
##     log(X5):log(X8) + log(X5):log(X9) + log(X6):log(X8) + log(X6):log(X9) + 
##     log(X7):log(X8) + log(X7):log(X9) + log(X8):log(X9)
## 
##                   Df Sum of Sq      RSS     AIC
## - log(X3):log(X8)  1 0.0000903 0.097990 -119.72
## - log(X3):log(X9)  1 0.0003856 0.098286 -119.63
## - log(X7):log(X8)  1 0.0008771 0.098777 -119.48
## - log(X2):log(X9)  1 0.0009602 0.098860 -119.46
## - log(X2):log(X8)  1 0.0012711 0.099171 -119.36
## - log(X4):log(X9)  1 0.0013307 0.099231 -119.34
## - log(X6):log(X8)  1 0.0013914 0.099291 -119.33
## - log(X7):log(X9)  1 0.0014236 0.099324 -119.32
## - log(X5):log(X8)  1 0.0014490 0.099349 -119.31
## - log(X5):log(X9)  1 0.0015560 0.099456 -119.28
## - log(X4):log(X8)  1 0.0016138 0.099514 -119.26
## - log(X6):log(X9)  1 0.0016209 0.099521 -119.26
## - log(X1):log(X9)  1 0.0025113 0.100411 -118.99
## - log(X1):log(X2)  1 0.0025483 0.100448 -118.98
## - log(X8):log(X9)  1 0.0025636 0.100464 -118.97
## - log(X1):log(X8)  1 0.0029919 0.100892 -118.85
## <none>                         0.097900 -117.75
## - log(X1):log(X3)  1 0.0093915 0.107292 -117.00
## 
## Step:  AIC=-119.72
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) + 
##     log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) + 
##     log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X8) + log(X2):log(X9) + 
##     log(X3):log(X9) + log(X4):log(X8) + log(X4):log(X9) + log(X5):log(X8) + 
##     log(X5):log(X9) + log(X6):log(X8) + log(X6):log(X9) + log(X7):log(X8) + 
##     log(X7):log(X9) + log(X8):log(X9)
## 
##                   Df Sum of Sq     RSS     AIC
## - log(X8):log(X9)  1  0.004379 0.10237 -120.41
## - log(X1):log(X2)  1  0.004551 0.10254 -120.36
## - log(X3):log(X9)  1  0.004640 0.10263 -120.33
## <none>                         0.09799 -119.72
## - log(X7):log(X8)  1  0.006852 0.10484 -119.69
## - log(X7):log(X9)  1  0.011626 0.10962 -118.36
## - log(X6):log(X8)  1  0.012314 0.11030 -118.17
## - log(X6):log(X9)  1  0.016183 0.11417 -117.14
## - log(X1):log(X9)  1  0.022498 0.12049 -115.52
## - log(X4):log(X9)  1  0.024248 0.12224 -115.09
## - log(X1):log(X8)  1  0.025269 0.12326 -114.84
## - log(X2):log(X9)  1  0.025677 0.12367 -114.74
## - log(X1):log(X3)  1  0.027330 0.12532 -114.34
## - log(X5):log(X9)  1  0.029018 0.12701 -113.94
## - log(X4):log(X8)  1  0.030440 0.12843 -113.61
## - log(X5):log(X8)  1  0.031030 0.12902 -113.47
## - log(X2):log(X8)  1  0.034946 0.13294 -112.57
## 
## Step:  AIC=-120.41
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) + 
##     log(X7) + log(X8) + log(X9) + log(X1):log(X2) + log(X1):log(X3) + 
##     log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X8) + log(X2):log(X9) + 
##     log(X3):log(X9) + log(X4):log(X8) + log(X4):log(X9) + log(X5):log(X8) + 
##     log(X5):log(X9) + log(X6):log(X8) + log(X6):log(X9) + log(X7):log(X8) + 
##     log(X7):log(X9)
## 
##                   Df Sum of Sq     RSS     AIC
## - log(X1):log(X2)  1 0.0019105 0.10428 -121.86
## - log(X7):log(X8)  1 0.0042768 0.10665 -121.18
## <none>                         0.10237 -120.41
## - log(X7):log(X9)  1 0.0085205 0.11089 -120.01
## - log(X6):log(X8)  1 0.0088118 0.11118 -119.93
## - log(X6):log(X9)  1 0.0122126 0.11458 -119.03
## - log(X1):log(X9)  1 0.0181216 0.12049 -117.52
## - log(X4):log(X9)  1 0.0203055 0.12268 -116.98
## - log(X3):log(X9)  1 0.0205839 0.12295 -116.91
## - log(X1):log(X8)  1 0.0209330 0.12330 -116.83
## - log(X2):log(X9)  1 0.0213148 0.12368 -116.74
## - log(X1):log(X3)  1 0.0237155 0.12609 -116.16
## - log(X5):log(X9)  1 0.0249321 0.12730 -115.87
## - log(X4):log(X8)  1 0.0267336 0.12910 -115.45
## - log(X5):log(X8)  1 0.0270663 0.12944 -115.37
## - log(X2):log(X8)  1 0.0305761 0.13295 -114.57
## 
## Step:  AIC=-121.86
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) + 
##     log(X7) + log(X8) + log(X9) + log(X1):log(X3) + log(X1):log(X8) + 
##     log(X1):log(X9) + log(X2):log(X8) + log(X2):log(X9) + log(X3):log(X9) + 
##     log(X4):log(X8) + log(X4):log(X9) + log(X5):log(X8) + log(X5):log(X9) + 
##     log(X6):log(X8) + log(X6):log(X9) + log(X7):log(X8) + log(X7):log(X9)
## 
##                   Df Sum of Sq     RSS     AIC
## - log(X7):log(X8)  1 0.0055209 0.10980 -122.31
## <none>                         0.10428 -121.86
## - log(X7):log(X9)  1 0.0100914 0.11437 -121.08
## - log(X6):log(X8)  1 0.0111603 0.11544 -120.81
## - log(X6):log(X9)  1 0.0144674 0.11875 -119.96
## - log(X2):log(X9)  1 0.0220327 0.12631 -118.11
## - log(X1):log(X9)  1 0.0224953 0.12677 -118.00
## - log(X3):log(X9)  1 0.0225020 0.12678 -118.00
## - log(X4):log(X9)  1 0.0227039 0.12698 -117.95
## - log(X1):log(X3)  1 0.0232574 0.12754 -117.82
## - log(X5):log(X9)  1 0.0244629 0.12874 -117.53
## - log(X5):log(X8)  1 0.0257774 0.13006 -117.23
## - log(X1):log(X8)  1 0.0262811 0.13056 -117.11
## - log(X4):log(X8)  1 0.0296051 0.13389 -116.36
## - log(X2):log(X8)  1 0.0312323 0.13551 -116.00
## 
## Step:  AIC=-122.31
## log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + log(X5) + log(X6) + 
##     log(X7) + log(X8) + log(X9) + log(X1):log(X3) + log(X1):log(X8) + 
##     log(X1):log(X9) + log(X2):log(X8) + log(X2):log(X9) + log(X3):log(X9) + 
##     log(X4):log(X8) + log(X4):log(X9) + log(X5):log(X8) + log(X5):log(X9) + 
##     log(X6):log(X8) + log(X6):log(X9) + log(X7):log(X9)
## 
##                   Df Sum of Sq     RSS      AIC
## <none>                         0.10980 -122.309
## - log(X3):log(X9)  1  0.020900 0.13070 -119.081
## - log(X1):log(X3)  1  0.021290 0.13109 -118.992
## - log(X7):log(X9)  1  0.023868 0.13367 -118.408
## - log(X6):log(X8)  1  0.030444 0.14024 -116.967
## - log(X5):log(X8)  1  0.032198 0.14200 -116.594
## - log(X1):log(X9)  1  0.034511 0.14431 -116.109
## - log(X5):log(X9)  1  0.036577 0.14638 -115.683
## - log(X6):log(X9)  1  0.057094 0.16689 -111.748
## - log(X1):log(X8)  1  0.058955 0.16876 -111.415
## - log(X2):log(X9)  1  0.067720 0.17752 -109.896
## - log(X4):log(X9)  1  0.086014 0.19581 -106.953
## - log(X2):log(X8)  1  0.117909 0.22771 -102.426
## - log(X4):log(X8)  1  0.199440 0.30924  -93.245
huxreg(model_wf_aic_log, model_wf_aic_all_log, model_wf_aic_log_inter, model_wf_aic_all_log_inter)
(1) (2) (3) (4)
(Intercept) 2.692 *** 0.571     -0.981     -16.027   
(0.445)    (3.360)    (1.520)    (13.947)  
X3 0.184 ***          0.493 ***        
(0.032)             (0.066)           
X4 0.109 ***          0.168 **         
(0.026)             (0.054)           
X6 -0.368 *                            
(0.146)                             
X7 4.085 **           2.515            
(1.213)             (1.258)           
X8 0.612 ***          0.586 ***        
(0.133)             (0.087)           
X9 -0.448 ***          -0.248 *          
(0.108)             (0.105)           
log(X1)          0.726 ***          1.158   
         (0.036)             (0.697)  
log(X3)          0.419 ***          1.666   
         (0.096)             (1.763)  
log(X5)          1.259              5.028   
         (0.796)             (3.473)  
log(X6)          -0.267 **           -0.158   
         (0.090)             (0.300)  
log(X8)          1.623 ***          44.919   
         (0.175)             (24.175)  
log(X9)          -1.375 ***          -37.066   
         (0.154)             (19.170)  
X1                   2.712 ***        
                  (0.330)           
X2                   -24.452 ***        
                  (5.631)           
X5                   0.039 *          
                  (0.016)           
X1:X3                   -0.416 ***        
                  (0.045)           
X1:X9                   -0.118 **         
                  (0.036)           
X2:X9                   4.643 **         
                  (1.449)           
X3:X8                   -0.051 **         
                  (0.014)           
X4:X8                   0.071 ***        
                  (0.015)           
X7:X9                   -0.995 *          
                  (0.344)           
log(X2)                            -0.733   
                           (0.338)  
log(X4)                            -2.394   
                           (2.963)  
log(X7)                            -0.408   
                           (0.518)  
log(X1):log(X3)                            0.823   
                           (0.707)  
log(X1):log(X8)                            1.242   
                           (0.641)  
log(X1):log(X9)                            -1.000   
                           (0.674)  
log(X2):log(X8)                            1.089 * 
                           (0.397)  
log(X2):log(X9)                            -0.710   
                           (0.341)  
log(X3):log(X9)                            0.411   
                           (0.356)  
log(X4):log(X8)                            -3.029 **
                           (0.849)  
log(X4):log(X9)                            2.174   
                           (0.929)  
log(X5):log(X8)                            -7.969   
                           (5.562)  
log(X5):log(X9)                            6.795   
                           (4.450)  
log(X6):log(X8)                            0.403   
                           (0.289)  
log(X6):log(X9)                            -0.506   
                           (0.265)  
log(X7):log(X9)                            0.464   
                           (0.376)  
N 30         30         30         30       
R2 0.947     0.986     0.994     0.998   
logLik -11.581     8.464     20.797     41.586   
AIC 39.161     -0.929     -9.594     -35.172   
*** p < 0.001; ** p < 0.01; * p < 0.05.
  • Mixed models
# Mixed regression 1
model_wf_mix1 <- lm(log(y) ~ (X1 + X3 + X4 )^2 + log(X2+ X5+X6 + X7) + log(X8) + log(X9), data=table_wf)
model_wf_aic_mix1 <- stepAIC(model_wf_mix1)
## Start:  AIC=-80.71
## log(y) ~ (X1 + X3 + X4)^2 + log(X2 + X5 + X6 + X7) + log(X8) + 
##     log(X9)
## 
## 
## Step:  AIC=-80.71
## log(y) ~ X1 + X3 + X4 + log(X2 + X5 + X6 + X7) + log(X8) + log(X9) + 
##     X1:X3 + X1:X4
## 
##                          Df Sum of Sq    RSS     AIC
## - log(X2 + X5 + X6 + X7)  1    0.0122 1.1294 -82.385
## <none>                                1.1172 -80.711
## - X1:X4                   1    0.2938 1.4110 -75.706
## - X1:X3                   1    1.8665 2.9837 -53.241
## - log(X9)                 1    3.2544 4.3716 -41.782
## - log(X8)                 1    3.4346 4.5518 -40.570
## 
## Step:  AIC=-82.38
## log(y) ~ X1 + X3 + X4 + log(X8) + log(X9) + X1:X3 + X1:X4
## 
##           Df Sum of Sq    RSS     AIC
## <none>                 1.1294 -82.385
## - X1:X4    1    0.3135 1.4429 -77.036
## - X1:X3    1    2.3929 3.5224 -50.262
## - log(X9)  1    3.9030 5.0324 -39.559
## - log(X8)  1    4.3068 5.4362 -37.243
# Mixed regression 2
model_wf_mix2 <- lm(log(y) ~ (X1 + X3 + X4 )^2 + log(X2)+log(X5)+log(X6)+log(X7) + log(X8) + log(X9), data=table_wf)
model_wf_aic_mix2 <- stepAIC(model_wf_mix2)
## Start:  AIC=-81.81
## log(y) ~ (X1 + X3 + X4)^2 + log(X2) + log(X5) + log(X6) + log(X7) + 
##     log(X8) + log(X9)
## 
## 
## Step:  AIC=-81.81
## log(y) ~ X1 + X3 + X4 + log(X2) + log(X5) + log(X6) + log(X7) + 
##     log(X8) + log(X9) + X1:X3 + X1:X4
## 
##           Df Sum of Sq    RSS     AIC
## - log(X6)  1    0.0002 0.8820 -83.802
## - log(X5)  1    0.0082 0.8900 -83.531
## - log(X2)  1    0.0112 0.8930 -83.431
## - log(X7)  1    0.0187 0.9005 -83.181
## - X1:X4    1    0.0379 0.9197 -82.547
## <none>                 0.8818 -81.810
## - X1:X3    1    0.3016 1.1834 -74.984
## - log(X9)  1    3.4852 4.3670 -35.813
## - log(X8)  1    3.6405 4.5223 -34.765
## 
## Step:  AIC=-83.8
## log(y) ~ X1 + X3 + X4 + log(X2) + log(X5) + log(X7) + log(X8) + 
##     log(X9) + X1:X3 + X1:X4
## 
##           Df Sum of Sq    RSS     AIC
## - log(X5)  1    0.0305 0.9125 -84.783
## - log(X2)  1    0.0553 0.9374 -83.977
## <none>                 0.8820 -83.802
## - log(X7)  1    0.1162 0.9982 -82.091
## - X1:X4    1    0.1973 1.0793 -79.746
## - X1:X3    1    1.9085 2.7905 -51.249
## - log(X9)  1    3.4883 4.3704 -37.791
## - log(X8)  1    3.6480 4.5300 -36.714
## 
## Step:  AIC=-84.78
## log(y) ~ X1 + X3 + X4 + log(X2) + log(X7) + log(X8) + log(X9) + 
##     X1:X3 + X1:X4
## 
##           Df Sum of Sq    RSS     AIC
## - log(X2)  1    0.0277 0.9402 -85.886
## <none>                 0.9125 -84.783
## - log(X7)  1    0.1121 1.0246 -83.308
## - X1:X4    1    0.1731 1.0856 -81.571
## - X1:X3    1    1.8806 2.7930 -53.222
## - log(X9)  1    3.6887 4.6012 -38.246
## - log(X8)  1    4.1243 5.0368 -35.533
## 
## Step:  AIC=-85.89
## log(y) ~ X1 + X3 + X4 + log(X7) + log(X8) + log(X9) + X1:X3 + 
##     X1:X4
## 
##           Df Sum of Sq    RSS     AIC
## <none>                 0.9402 -85.886
## - log(X7)  1    0.1892 1.1294 -82.385
## - X1:X4    1    0.2211 1.1613 -81.549
## - X1:X3    1    2.5494 3.4896 -48.542
## - log(X9)  1    4.0912 5.0314 -37.565
## - log(X8)  1    4.4859 5.4261 -35.300
# Mixed regression 3
model_wf_mix3 <- lm(log(y) ~ (X1 + X3 + X4 )^2 + (log(X2)+log(X5)+log(X6)+log(X7))^2 + log(X8) + log(X9), data=table_wf)
model_wf_aic_mix3 <- stepAIC(model_wf_mix3)
## Start:  AIC=-81.81
## log(y) ~ (X1 + X3 + X4)^2 + (log(X2) + log(X5) + log(X6) + log(X7))^2 + 
##     log(X8) + log(X9)
## 
## 
## Step:  AIC=-81.81
## log(y) ~ X1 + X3 + X4 + log(X2) + log(X5) + log(X6) + log(X7) + 
##     log(X8) + log(X9) + X1:X3 + X1:X4 + X3:X4 + log(X2):log(X5) + 
##     log(X2):log(X6) + log(X2):log(X7) + log(X5):log(X6) + log(X5):log(X7)
## 
## 
## Step:  AIC=-81.81
## log(y) ~ X1 + X3 + X4 + log(X2) + log(X5) + log(X6) + log(X7) + 
##     log(X8) + log(X9) + X1:X3 + X1:X4 + X3:X4 + log(X2):log(X5) + 
##     log(X2):log(X6) + log(X2):log(X7) + log(X5):log(X6)
## 
## 
## Step:  AIC=-81.81
## log(y) ~ X1 + X3 + X4 + log(X2) + log(X5) + log(X6) + log(X7) + 
##     log(X8) + log(X9) + X1:X3 + X1:X4 + X3:X4 + log(X2):log(X5) + 
##     log(X2):log(X6) + log(X2):log(X7)
## 
## 
## Step:  AIC=-81.81
## log(y) ~ X1 + X3 + X4 + log(X2) + log(X5) + log(X6) + log(X7) + 
##     log(X8) + log(X9) + X1:X3 + X1:X4 + X3:X4 + log(X2):log(X5) + 
##     log(X2):log(X6)
## 
## 
## Step:  AIC=-81.81
## log(y) ~ X1 + X3 + X4 + log(X2) + log(X5) + log(X6) + log(X7) + 
##     log(X8) + log(X9) + X1:X3 + X1:X4 + X3:X4 + log(X2):log(X5)
## 
## 
## Step:  AIC=-81.81
## log(y) ~ X1 + X3 + X4 + log(X2) + log(X5) + log(X6) + log(X7) + 
##     log(X8) + log(X9) + X1:X3 + X1:X4 + X3:X4
## 
## 
## Step:  AIC=-81.81
## log(y) ~ X1 + X3 + X4 + log(X2) + log(X5) + log(X6) + log(X7) + 
##     log(X8) + log(X9) + X1:X3 + X1:X4
## 
##           Df Sum of Sq    RSS     AIC
## - log(X6)  1    0.0002 0.8820 -83.802
## - log(X5)  1    0.0082 0.8900 -83.531
## - log(X2)  1    0.0112 0.8930 -83.431
## - log(X7)  1    0.0187 0.9005 -83.181
## - X1:X4    1    0.0379 0.9197 -82.547
## <none>                 0.8818 -81.810
## - X1:X3    1    0.3016 1.1834 -74.984
## - log(X9)  1    3.4852 4.3670 -35.813
## - log(X8)  1    3.6405 4.5223 -34.765
## 
## Step:  AIC=-83.8
## log(y) ~ X1 + X3 + X4 + log(X2) + log(X5) + log(X7) + log(X8) + 
##     log(X9) + X1:X3 + X1:X4
## 
##           Df Sum of Sq    RSS     AIC
## - log(X5)  1    0.0305 0.9125 -84.783
## - log(X2)  1    0.0553 0.9374 -83.977
## <none>                 0.8820 -83.802
## - log(X7)  1    0.1162 0.9982 -82.091
## - X1:X4    1    0.1973 1.0793 -79.746
## - X1:X3    1    1.9085 2.7905 -51.249
## - log(X9)  1    3.4883 4.3704 -37.791
## - log(X8)  1    3.6480 4.5300 -36.714
## 
## Step:  AIC=-84.78
## log(y) ~ X1 + X3 + X4 + log(X2) + log(X7) + log(X8) + log(X9) + 
##     X1:X3 + X1:X4
## 
##           Df Sum of Sq    RSS     AIC
## - log(X2)  1    0.0277 0.9402 -85.886
## <none>                 0.9125 -84.783
## - log(X7)  1    0.1121 1.0246 -83.308
## - X1:X4    1    0.1731 1.0856 -81.571
## - X1:X3    1    1.8806 2.7930 -53.222
## - log(X9)  1    3.6887 4.6012 -38.246
## - log(X8)  1    4.1243 5.0368 -35.533
## 
## Step:  AIC=-85.89
## log(y) ~ X1 + X3 + X4 + log(X7) + log(X8) + log(X9) + X1:X3 + 
##     X1:X4
## 
##           Df Sum of Sq    RSS     AIC
## <none>                 0.9402 -85.886
## - log(X7)  1    0.1892 1.1294 -82.385
## - X1:X4    1    0.2211 1.1613 -81.549
## - X1:X3    1    2.5494 3.4896 -48.542
## - log(X9)  1    4.0912 5.0314 -37.565
## - log(X8)  1    4.4859 5.4261 -35.300
huxreg(model_wf_aic_mix1, model_wf_aic_mix2, model_wf_aic_mix3)
(1) (2) (3)
(Intercept) 2.598 *** 2.044 *** 2.044 ***
(0.228)    (0.343)    (0.343)   
X1 1.290 *** 1.459 *** 1.459 ***
(0.232)    (0.232)    (0.232)   
X3 0.296 *** 0.275 *** 0.275 ***
(0.040)    (0.039)    (0.039)   
X4 0.391 *** 0.423 *** 0.423 ***
(0.042)    (0.042)    (0.042)   
log(X8) 1.575 *** 1.628 *** 1.628 ***
(0.172)    (0.163)    (0.163)   
log(X9) -1.345 *** -1.405 *** -1.405 ***
(0.154)    (0.147)    (0.147)   
X1:X3 -0.381 *** -0.398 *** -0.398 ***
(0.056)    (0.053)    (0.053)   
X1:X4 0.031 *   0.026 *   0.026 *  
(0.012)    (0.012)    (0.012)   
log(X7)          -0.428     -0.428    
         (0.208)    (0.208)   
N 30         30         30        
R2 0.984     0.987     0.987    
logLik 6.624     9.375     9.375    
AIC 4.752     1.250     1.250    
*** p < 0.001; ** p < 0.01; * p < 0.05.

Additional Steps

  • step both
# Stepwise Regression based on p values for full model#
k <- ols_step_both_p(model_wf_full_log)
## Stepwise Selection Method   
## ---------------------------
## 
## Candidate Terms: 
## 
## 1. X1 
## 2. X2 
## 3. X3 
## 4. X4 
## 5. X5 
## 6. X6 
## 7. X7 
## 8. X8 
## 9. X9 
## 
## We are selecting variables based on p value...
## 
## Variables Entered/Removed: 
## 
## - X4 added 
## - X3 added 
## - X7 added 
## 
## No more variables to be added/removed.
## 
## 
## Final Model Output 
## ------------------
## 
##                         Model Summary                         
## -------------------------------------------------------------
## R                       0.944       RMSE               0.549 
## R-Squared               0.890       Coef. Var          8.618 
## Adj. R-Squared          0.878       MSE                0.301 
## Pred R-Squared          0.854       MAE                0.414 
## -------------------------------------------------------------
##  RMSE: Root Mean Square Error 
##  MSE: Mean Square Error 
##  MAE: Mean Absolute Error 
## 
##                                ANOVA                                
## -------------------------------------------------------------------
##                Sum of                                              
##               Squares        DF    Mean Square      F         Sig. 
## -------------------------------------------------------------------
## Regression     63.565         3         21.188    70.378    0.0000 
## Residual        7.828        26          0.301                     
## Total          71.393        29                                    
## -------------------------------------------------------------------
## 
##                                  Parameter Estimates                                  
## -------------------------------------------------------------------------------------
##       model     Beta    Std. Error    Std. Beta      t       Sig      lower    upper 
## -------------------------------------------------------------------------------------
## (Intercept)    2.872         0.547                 5.254    0.000     1.748    3.995 
##          X4    0.122         0.033        0.559    3.730    0.001     0.055    0.189 
##          X3    0.168         0.040        0.435    4.165    0.000     0.085    0.251 
##          X7    3.106         1.537        0.309    2.021    0.054    -0.053    6.266 
## -------------------------------------------------------------------------------------
 k
## 
##                              Stepwise Selection Summary                              
## ------------------------------------------------------------------------------------
##                      Added/                   Adj.                                      
## Step    Variable    Removed     R-Square    R-Square     C(p)        AIC       RMSE     
## ------------------------------------------------------------------------------------
##    1       X4       addition       0.803       0.796    48.8550    68.4060    0.7087    
##    2       X3       addition       0.873       0.864    24.2130    57.2082    0.5792    
##    3       X7       addition       0.890       0.878    19.6670    54.8305    0.5487    
## ------------------------------------------------------------------------------------
# plot(k)

# Stepwise AIC Regression for full model#
k<- ols_step_both_aic(model_wf_full_log)
## Stepwise Selection Method 
## -------------------------
## 
## Candidate Terms: 
## 
## 1 . X1 
## 2 . X2 
## 3 . X3 
## 4 . X4 
## 5 . X5 
## 6 . X6 
## 7 . X7 
## 8 . X8 
## 9 . X9 
## 
## 
## Variables Entered/Removed: 
## 
## - X4 added 
## - X3 added 
## - X7 added 
## - X8 added 
## - X9 added 
## - X6 added 
## 
## No more variables to be added or removed.
 k
## 
## 
##                               Stepwise Summary                              
## --------------------------------------------------------------------------
## Variable     Method      AIC       RSS      Sum Sq     R-Sq      Adj. R-Sq 
## --------------------------------------------------------------------------
## X4          addition    68.406    14.063    57.330    0.80302      0.79599 
## X3          addition    57.208     9.057    62.335    0.87313      0.86373 
## X7          addition    54.830     7.828    63.565    0.89036      0.87771 
## X8          addition    54.522     7.248    64.144    0.89848      0.88223 
## X9          addition    44.504     4.856    66.537    0.93199      0.91782 
## X6          addition    39.161     3.801    67.591    0.94675      0.93286 
## --------------------------------------------------------------------------
# plot(k)

# Stepwise Regression based on p values for all log model #
k <- ols_step_both_p(model_wf_all_log)
## Stepwise Selection Method   
## ---------------------------
## 
## Candidate Terms: 
## 
## 1. log(X1) 
## 2. log(X2) 
## 3. log(X3) 
## 4. log(X4) 
## 5. log(X5) 
## 6. log(X6) 
## 7. log(X7) 
## 8. log(X8) 
## 9. log(X9) 
## 
## We are selecting variables based on p value...
## 
## Variables Entered/Removed: 
## 
## - log(X4) added 
## 
## No more variables to be added/removed.
## 
## 
## Final Model Output 
## ------------------
## 
##                         Model Summary                         
## -------------------------------------------------------------
## R                       0.954       RMSE               0.479 
## R-Squared               0.910       Coef. Var          7.526 
## Adj. R-Squared          0.907       MSE                0.230 
## Pred R-Squared          0.896       MAE                0.353 
## -------------------------------------------------------------
##  RMSE: Root Mean Square Error 
##  MSE: Mean Square Error 
##  MAE: Mean Absolute Error 
## 
##                                ANOVA                                 
## --------------------------------------------------------------------
##                Sum of                                               
##               Squares        DF    Mean Square       F         Sig. 
## --------------------------------------------------------------------
## Regression     64.964         1         64.964    282.937    0.0000 
## Residual        6.429        28          0.230                      
## Total          71.393        29                                     
## --------------------------------------------------------------------
## 
##                                  Parameter Estimates                                  
## -------------------------------------------------------------------------------------
##       model     Beta    Std. Error    Std. Beta      t        Sig     lower    upper 
## -------------------------------------------------------------------------------------
## (Intercept)    4.189         0.156                 26.803    0.000    3.868    4.509 
##     log(X4)    1.259         0.075        0.954    16.821    0.000    1.106    1.413 
## -------------------------------------------------------------------------------------
k
## 
##                              Stepwise Selection Summary                               
## -------------------------------------------------------------------------------------
##                      Added/                   Adj.                                       
## Step    Variable    Removed     R-Square    R-Square      C(p)        AIC       RMSE     
## -------------------------------------------------------------------------------------
##    1    log(X4)     addition       0.910       0.907    113.3920    44.9246    0.4792    
## -------------------------------------------------------------------------------------
#plot(k)

# Stepwise AIC Regression for all log model #
k <- ols_step_both_aic(model_wf_all_log)
## Stepwise Selection Method 
## -------------------------
## 
## Candidate Terms: 
## 
## 1 . log(X1) 
## 2 . log(X2) 
## 3 . log(X3) 
## 4 . log(X4) 
## 5 . log(X5) 
## 6 . log(X6) 
## 7 . log(X7) 
## 8 . log(X8) 
## 9 . log(X9) 
## 
## 
## Variables Entered/Removed: 
## 
## - log(X4) added 
## - log(X8) added 
## - log(X9) added 
## - log(X6) added 
## - log(X1) added 
## - log(X3) added 
## 
## No more variables to be added or removed.
k
## 
## 
##                              Stepwise Summary                              
## -------------------------------------------------------------------------
## Variable     Method      AIC       RSS     Sum Sq     R-Sq      Adj. R-Sq 
## -------------------------------------------------------------------------
## log(X4)     addition    44.925    6.429    64.964    0.90995      0.90673 
## log(X8)     addition    44.798    5.989    65.404    0.91611      0.90990 
## log(X9)     addition    14.099    2.014    69.379    0.97179      0.96854 
## log(X6)     addition     4.579    1.372    70.021    0.98079      0.97771 
## log(X1)     addition     2.166    1.184    70.209    0.98342      0.97996 
## log(X3)     addition     0.009    1.031    70.362    0.98556      0.98180 
## -------------------------------------------------------------------------
# plot(k)

# Stepwise Regression based on p values for all log model #
k <- ols_step_both_p(model_wf_full_log_inter)
## Stepwise Selection Method   
## ---------------------------
## 
## Candidate Terms: 
## 
## 1. X1 
## 2. X2 
## 3. X3 
## 4. X4 
## 5. X5 
## 6. X6 
## 7. X7 
## 8. X8 
## 9. X9 
## 10. X1:X2 
## 11. X1:X3 
## 12. X1:X4 
## 13. X1:X5 
## 14. X1:X6 
## 15. X1:X7 
## 16. X1:X8 
## 17. X1:X9 
## 18. X2:X3 
## 19. X2:X4 
## 20. X2:X5 
## 21. X2:X6 
## 22. X2:X7 
## 23. X2:X8 
## 24. X2:X9 
## 25. X3:X4 
## 26. X3:X5 
## 27. X3:X6 
## 28. X3:X7 
## 29. X3:X8 
## 30. X3:X9 
## 31. X4:X5 
## 32. X4:X6 
## 33. X4:X7 
## 34. X4:X8 
## 35. X4:X9 
## 36. X5:X6 
## 37. X5:X7 
## 38. X5:X8 
## 39. X5:X9 
## 40. X6:X7 
## 41. X6:X8 
## 42. X6:X9 
## 43. X7:X8 
## 44. X7:X9 
## 45. X8:X9 
## 
## We are selecting variables based on p value...
## 
## Variables Entered/Removed: 
## 
## - X3:X7 added 
## - X6 added 
## - X8 added 
## - X9 added 
## - X3 added 
## - X5 added 
## - X1 added 
## - X3 added 
## - X7 added 
## - X9 added 
## - X2 added 
## - X1 added 
## - X4 added 
## - X8 added 
## - X3:X4 added 
## - X3:X7 added 
## - X3:X5 added 
## - X6 added 
## - X1:X8 added 
## - X5 added 
## - X6:X9 added 
## - X2 added 
## - X6:X8 added 
## - X1:X4 added 
## - X7 added 
## 
## No more variables to be added/removed.
## 
## 
## Final Model Output 
## ------------------
## 
##                         Model Summary                         
## -------------------------------------------------------------
## R                       0.991       RMSE               0.237 
## R-Squared               0.983       Coef. Var          3.727 
## Adj. R-Squared          0.977       MSE                0.056 
## Pred R-Squared          0.961       MAE                0.153 
## -------------------------------------------------------------
##  RMSE: Root Mean Square Error 
##  MSE: Mean Square Error 
##  MAE: Mean Absolute Error 
## 
##                                ANOVA                                 
## --------------------------------------------------------------------
##                Sum of                                               
##               Squares        DF    Mean Square       F         Sig. 
## --------------------------------------------------------------------
## Regression     70.154         7         10.022    178.035    0.0000 
## Residual        1.238        22          0.056                      
## Total          71.393        29                                     
## --------------------------------------------------------------------
## 
##                                   Parameter Estimates                                    
## ----------------------------------------------------------------------------------------
##       model      Beta    Std. Error    Std. Beta      t        Sig      lower     upper 
## ----------------------------------------------------------------------------------------
## (Intercept)     2.602         0.256                 10.163    0.000     2.071     3.134 
##          X4     0.411         0.036        1.881    11.319    0.000     0.336     0.486 
##       X4:X3    -0.031         0.005       -1.207    -6.568    0.000    -0.041    -0.022 
##       X3:X5     0.005         0.001        0.860     7.218    0.000     0.004     0.007 
##       X1:X8     0.048         0.010        0.313     4.597    0.000     0.026     0.070 
##       X6:X9    -0.303         0.045       -0.584    -6.769    0.000    -0.395    -0.210 
##       X8:X6     0.297         0.054        0.486     5.489    0.000     0.185     0.409 
##       X4:X1    -0.012         0.003       -0.398    -3.689    0.001    -0.019    -0.005 
## ----------------------------------------------------------------------------------------
k
## 
##                              Stepwise Selection Summary                              
## ------------------------------------------------------------------------------------
##                      Added/                   Adj.                                      
## Step    Variable    Removed     R-Square    R-Square     C(p)        AIC       RMSE     
## ------------------------------------------------------------------------------------
##    1     X3:X7      addition       0.847       0.842    92.7180    60.7989    0.6243    
##    2       X6       addition       0.862       0.851    83.4120    59.7966    0.6047    
##    3       X8       addition       0.873       0.859    76.3490    59.1522    0.5897    
##    4       X9       addition       0.914       0.900    46.6570    49.4831    0.4951    
##    5       X3       addition       0.915       0.897    48.3380    51.3395    0.5041    
##    6       X5       addition       0.927       0.908    40.9360    48.7543    0.4770    
##    7       X1       addition       0.950       0.934    24.6230    39.1116    0.4017    
##    8       X3       removal        0.934       0.917    34.9220    45.4055    0.4511    
##    9       X7       addition       0.953       0.938    22.6940    37.5750    0.3916    
##   10       X9       removal        0.918       0.896    47.8030    52.1707    0.5050    
##   11       X2       addition       0.919       0.893    49.0940    53.8354    0.5134    
##   12       X1       removal        0.893       0.865    66.9630    60.0481    0.5758    
##   13       X4       addition       0.915       0.887    52.3760    55.3564    0.5266    
##   14       X8       removal        0.906       0.881    57.1400    56.2676    0.5407    
##   15     X3:X4      addition       0.934       0.913    37.2840    47.6179    0.4629    
##   16     X3:X7      removal        0.861       0.825    91.9210    67.9385    0.6568    
##   17     X3:X5      addition       0.934       0.913    37.3040    47.6297    0.4630    
##   18       X6       removal        0.934       0.917    35.3190    45.6383    0.4529    
##   19     X1:X8      addition       0.946       0.929    27.6400    41.3686    0.4171    
##   20       X5       removal        0.940       0.924    30.8160    42.8830    0.4326    
##   21     X6:X9      addition       0.964       0.953    13.8780    29.3316    0.3413    
##   22       X2       removal        0.964       0.955    11.8800    27.3333    0.3338    
##   23     X6:X8      addition       0.979       0.972     2.2430    13.1258    0.2605    
##   24     X1:X4      addition       0.984       0.977     0.7850     7.9454    0.2366    
##   25       X7       removal        0.983       0.977    -0.5280     7.5158    0.2373    
## ------------------------------------------------------------------------------------
# plot(k)

# Stepwise AIC Regression for all log model #
k <- ols_step_both_aic(model_wf_full_log_inter)
## Stepwise Selection Method 
## -------------------------
## 
## Candidate Terms: 
## 
## 1 . X1 
## 2 . X2 
## 3 . X3 
## 4 . X4 
## 5 . X5 
## 6 . X6 
## 7 . X7 
## 8 . X8 
## 9 . X9 
## 10 . X1:X2 
## 11 . X1:X3 
## 12 . X1:X4 
## 13 . X1:X5 
## 14 . X1:X6 
## 15 . X1:X7 
## 16 . X1:X8 
## 17 . X1:X9 
## 18 . X2:X3 
## 19 . X2:X4 
## 20 . X2:X5 
## 21 . X2:X6 
## 22 . X2:X7 
## 23 . X2:X8 
## 24 . X2:X9 
## 25 . X3:X4 
## 26 . X3:X5 
## 27 . X3:X6 
## 28 . X3:X7 
## 29 . X3:X8 
## 30 . X3:X9 
## 31 . X4:X5 
## 32 . X4:X6 
## 33 . X4:X7 
## 34 . X4:X8 
## 35 . X4:X9 
## 36 . X5:X6 
## 37 . X5:X7 
## 38 . X5:X8 
## 39 . X5:X9 
## 40 . X6:X7 
## 41 . X6:X8 
## 42 . X6:X9 
## 43 . X7:X8 
## 44 . X7:X9 
## 45 . X8:X9 
## 
## 
## Variables Entered/Removed: 
## 
## - X3:X7 added 
## - X4:X5 added 
## - X3:X5 added 
## - X3:X4 added 
## - X3:X7 removed 
## - X1:X8 added 
## - X9 added 
## - X8 added 
## - X6:X9 added 
## - X6:X7 added 
## - X3:X9 added 
## - X3:X8 added 
## 
## No more variables to be added or removed.
k
## 
## 
##                               Stepwise Summary                              
## --------------------------------------------------------------------------
## Variable     Method      AIC       RSS      Sum Sq     R-Sq      Adj. R-Sq 
## --------------------------------------------------------------------------
## X3:X7       addition    60.799    10.913    60.480    0.84714      0.84168 
## X4:X5       addition    54.665     8.321    63.071    0.88344      0.87481 
## X3:X5       addition    50.611     6.801    64.592    0.90474      0.89375 
## X3:X4       addition    46.641     5.574    65.819    0.92193      0.90944 
## X3:X7       removal     44.948     5.631    65.762    0.92113      0.91203 
## X1:X8       addition    37.518     4.112    67.280    0.94240      0.93319 
## X9          addition    24.336     2.479    68.914    0.96528      0.95804 
## X8          addition     6.182     1.266    70.126    0.98226      0.97764 
## X6:X9       addition     3.316     1.077    70.316    0.98492      0.98012 
## X6:X7       addition    -4.379     0.779    70.613    0.98908      0.98493 
## X3:X9       addition    -4.884     0.717    70.676    0.98996      0.98544 
## X3:X8       addition    -5.046     0.667    70.725    0.99066      0.98574 
## --------------------------------------------------------------------------
# plot(k)

# Stepwise Regression based on p values for all log model #
k <- ols_step_both_p(model_wf_all_log_inter)
## Stepwise Selection Method   
## ---------------------------
## 
## Candidate Terms: 
## 
## 1. log(X1) 
## 2. log(X2) 
## 3. log(X3) 
## 4. log(X4) 
## 5. log(X5) 
## 6. log(X6) 
## 7. log(X7) 
## 8. log(X8) 
## 9. log(X9) 
## 10. log(X1):log(X2) 
## 11. log(X1):log(X3) 
## 12. log(X1):log(X4) 
## 13. log(X1):log(X5) 
## 14. log(X1):log(X6) 
## 15. log(X1):log(X7) 
## 16. log(X1):log(X8) 
## 17. log(X1):log(X9) 
## 18. log(X2):log(X3) 
## 19. log(X2):log(X4) 
## 20. log(X2):log(X5) 
## 21. log(X2):log(X6) 
## 22. log(X2):log(X7) 
## 23. log(X2):log(X8) 
## 24. log(X2):log(X9) 
## 25. log(X3):log(X4) 
## 26. log(X3):log(X5) 
## 27. log(X3):log(X6) 
## 28. log(X3):log(X7) 
## 29. log(X3):log(X8) 
## 30. log(X3):log(X9) 
## 31. log(X4):log(X5) 
## 32. log(X4):log(X6) 
## 33. log(X4):log(X7) 
## 34. log(X4):log(X8) 
## 35. log(X4):log(X9) 
## 36. log(X5):log(X6) 
## 37. log(X5):log(X7) 
## 38. log(X5):log(X8) 
## 39. log(X5):log(X9) 
## 40. log(X6):log(X7) 
## 41. log(X6):log(X8) 
## 42. log(X6):log(X9) 
## 43. log(X7):log(X8) 
## 44. log(X7):log(X9) 
## 45. log(X8):log(X9) 
## 
## We are selecting variables based on p value...
## 
## Variables Entered/Removed: 
## 
## - log(X1):log(X2) added 
## - log(X8) added 
## - log(X9) added 
## - log(X6) added 
## - log(X7) added 
## - log(X6) added 
## - log(X2) added 
## - log(X9) added 
## - log(X5) added 
## - log(X8) added 
## - log(X3) added 
## - log(X1):log(X2) added 
## - log(X1) added 
## - log(X7) added 
## - log(X1):log(X9) added 
## - log(X2) added 
## - log(X4) added 
## - log(X1):log(X9) added 
## - log(X4):log(X8) added 
## - log(X4) added 
## - log(X3):log(X9) added 
## - log(X2):log(X8) added 
## - log(X6):log(X7) added 
## - log(X2):log(X9) added 
## 
## No more variables to be added/removed.
## 
## 
## Final Model Output 
## ------------------
## 
##                         Model Summary                         
## -------------------------------------------------------------
## R                       0.996       RMSE               0.171 
## R-Squared               0.991       Coef. Var          2.685 
## Adj. R-Squared          0.988       MSE                0.029 
## Pred R-Squared          0.981       MAE                0.109 
## -------------------------------------------------------------
##  RMSE: Root Mean Square Error 
##  MSE: Mean Square Error 
##  MAE: Mean Absolute Error 
## 
##                                ANOVA                                 
## --------------------------------------------------------------------
##                Sum of                                               
##               Squares        DF    Mean Square       F         Sig. 
## --------------------------------------------------------------------
## Regression     70.779         8          8.847    302.669    0.0000 
## Residual        0.614        21          0.029                      
## Total          71.393        29                                     
## --------------------------------------------------------------------
## 
##                                     Parameter Estimates                                      
## --------------------------------------------------------------------------------------------
##           model      Beta    Std. Error    Std. Beta      t        Sig      lower     upper 
## --------------------------------------------------------------------------------------------
##     (Intercept)    -3.681         2.633                 -1.398    0.177    -9.156     1.794 
##         log(X5)     2.133         0.620        0.141     3.438    0.002     0.843     3.423 
##         log(X3)     0.690         0.095        0.228     7.286    0.000     0.493     0.887 
##         log(X1)     0.603         0.048        0.803    12.443    0.000     0.502     0.704 
## log(X4):log(X8)     0.358         0.065        0.371     5.490    0.000     0.222     0.494 
## log(X3):log(X9)    -0.379         0.106       -0.297    -3.565    0.002    -0.601    -0.158 
## log(X8):log(X2)    -0.219         0.040       -0.280    -5.531    0.000    -0.301    -0.136 
## log(X6):log(X7)     0.185         0.053        0.077     3.460    0.002     0.074     0.296 
## log(X9):log(X2)     0.133         0.052        0.203     2.554    0.019     0.025     0.241 
## --------------------------------------------------------------------------------------------
k
## 
##                                  Stepwise Selection Summary                                   
## ---------------------------------------------------------------------------------------------
##                             Added/                   Adj.                                        
## Step       Variable        Removed     R-Square    R-Square      C(p)        AIC        RMSE     
## ---------------------------------------------------------------------------------------------
##    1    log(X1):log(X2)    addition       0.915       0.912    160.4600     43.2742    0.4662    
##    2        log(X8)        addition       0.920       0.914    151.7000     43.4911    0.4608    
##    3        log(X9)        addition       0.970       0.966     44.6280     16.4013    0.2892    
##    4        log(X6)        addition       0.972       0.967     41.5170     16.0073    0.2834    
##    5        log(X7)        addition       0.972       0.966     43.4490     17.9741    0.2891    
##    6        log(X6)        removal        0.970       0.965     46.4730     18.3319    0.2946    
##    7        log(X2)        addition       0.975       0.970     36.8870     14.5859    0.2732    
##    8        log(X9)        removal        0.926       0.915    141.2310     44.9129    0.4588    
##    9        log(X5)        addition       0.929       0.914    137.0590     45.7420    0.4592    
##   10        log(X8)        removal        0.924       0.912    145.3660     45.6727    0.4646    
##   11        log(X3)        addition       0.927       0.912    140.6940     46.4372    0.4645    
##   12    log(X1):log(X2)    removal        0.892       0.875    215.3970     56.2660    0.5543    
##   13        log(X1)        addition       0.933       0.919    128.7640     44.0925    0.4467    
##   14        log(X7)        removal        0.933       0.922    126.8600     42.1123    0.4378    
##   15    log(X1):log(X9)    addition       0.950       0.940     91.2400     35.2341    0.3854    
##   16        log(X2)        removal        0.949       0.941     90.5010     33.5785    0.3798    
##   17        log(X4)        addition       0.950       0.939     92.4090     35.5536    0.3875    
##   18    log(X1):log(X9)    removal        0.932       0.921    128.0910     42.3625    0.4397    
##   19    log(X4):log(X8)    addition       0.941       0.929    111.3120     40.2947    0.4193    
##   20        log(X4)        removal        0.940       0.931    110.3630     38.5375    0.4125    
##   21    log(X3):log(X9)    addition       0.975       0.970     35.7360     13.9503    0.2703    
##   22    log(X2):log(X8)    addition       0.984       0.980     18.4730      2.6333    0.2212    
##   23    log(X6):log(X7)    addition       0.989       0.985     10.6510     -5.4275    0.1912    
##   24    log(X2):log(X9)    addition       0.991       0.988      6.8110    -11.5396    0.1710    
## ---------------------------------------------------------------------------------------------
# plot(k)

# Stepwise AIC Regression for all log model #
k <- ols_step_both_aic(model_wf_all_log_inter)
## Stepwise Selection Method 
## -------------------------
## 
## Candidate Terms: 
## 
## 1 . log(X1) 
## 2 . log(X2) 
## 3 . log(X3) 
## 4 . log(X4) 
## 5 . log(X5) 
## 6 . log(X6) 
## 7 . log(X7) 
## 8 . log(X8) 
## 9 . log(X9) 
## 10 . log(X1):log(X2) 
## 11 . log(X1):log(X3) 
## 12 . log(X1):log(X4) 
## 13 . log(X1):log(X5) 
## 14 . log(X1):log(X6) 
## 15 . log(X1):log(X7) 
## 16 . log(X1):log(X8) 
## 17 . log(X1):log(X9) 
## 18 . log(X2):log(X3) 
## 19 . log(X2):log(X4) 
## 20 . log(X2):log(X5) 
## 21 . log(X2):log(X6) 
## 22 . log(X2):log(X7) 
## 23 . log(X2):log(X8) 
## 24 . log(X2):log(X9) 
## 25 . log(X3):log(X4) 
## 26 . log(X3):log(X5) 
## 27 . log(X3):log(X6) 
## 28 . log(X3):log(X7) 
## 29 . log(X3):log(X8) 
## 30 . log(X3):log(X9) 
## 31 . log(X4):log(X5) 
## 32 . log(X4):log(X6) 
## 33 . log(X4):log(X7) 
## 34 . log(X4):log(X8) 
## 35 . log(X4):log(X9) 
## 36 . log(X5):log(X6) 
## 37 . log(X5):log(X7) 
## 38 . log(X5):log(X8) 
## 39 . log(X5):log(X9) 
## 40 . log(X6):log(X7) 
## 41 . log(X6):log(X8) 
## 42 . log(X6):log(X9) 
## 43 . log(X7):log(X8) 
## 44 . log(X7):log(X9) 
## 45 . log(X8):log(X9) 
## 
## 
## Variables Entered/Removed: 
## 
## - log(X1):log(X2) added 
## - log(X1):log(X9) added 
## - log(X1):log(X4) added 
## - log(X5):log(X8) added 
## - log(X3):log(X9) added 
## - log(X1):log(X5) added 
## - log(X4):log(X8) added 
## - log(X1):log(X9) removed 
## - log(X6) added 
## - log(X9) added 
## - log(X3):log(X8) added 
## - log(X3):log(X9) removed 
## - log(X4):log(X9) added 
## - log(X2):log(X4) added 
## - log(X7):log(X9) added 
## - log(X7):log(X8) added 
## 
## No more variables to be added or removed.
k
## 
## 
##                                  Stepwise Summary                                  
## ---------------------------------------------------------------------------------
## Variable            Method       AIC       RSS     Sum Sq     R-Sq      Adj. R-Sq 
## ---------------------------------------------------------------------------------
## log(X1):log(X2)    addition     43.274    6.085    65.308    0.91477      0.91173 
## log(X1):log(X9)    addition     40.858    5.252    66.141    0.92644      0.92099 
## log(X1):log(X4)    addition     36.333    4.225    67.167    0.94082      0.93399 
## log(X5):log(X8)    addition     33.745    3.626    67.766    0.94921      0.94108 
## log(X3):log(X9)    addition      9.187    1.496    69.896    0.97904      0.97468 
## log(X1):log(X5)    addition     -1.660    0.975    70.418    0.98634      0.98278 
## log(X4):log(X8)    addition     -3.960    0.845    70.548    0.98817      0.98440 
## log(X1):log(X9)    removal      -4.665    0.882    70.510    0.98765      0.98442 
## log(X6)            addition     -7.138    0.760    70.633    0.98936      0.98597 
## log(X9)            addition    -17.032    0.511    70.881    0.99284      0.99011 
## log(X3):log(X8)    addition    -17.687    0.468    70.925    0.99345      0.99050 
## log(X3):log(X9)    removal     -19.540    0.470    70.922    0.99341      0.99091 
## log(X4):log(X9)    addition    -21.843    0.407    70.985    0.99429      0.99173 
## log(X2):log(X4)    addition    -22.522    0.373    71.020    0.99478      0.99204 
## log(X7):log(X9)    addition    -22.985    0.343    71.049    0.99519      0.99226 
## log(X7):log(X8)    addition    -28.798    0.264    71.128    0.99630      0.99368 
## ---------------------------------------------------------------------------------
# plot(k)

# Stepwise Regression based on p values for all log model #
k <- ols_step_both_p(model_wf_mix2 )
## Stepwise Selection Method   
## ---------------------------
## 
## Candidate Terms: 
## 
## 1. X1 
## 2. X3 
## 3. X4 
## 4. log(X2) 
## 5. log(X5) 
## 6. log(X6) 
## 7. log(X7) 
## 8. log(X8) 
## 9. log(X9) 
## 10. X1:X3 
## 11. X1:X4 
## 12. X3:X4 
## 
## We are selecting variables based on p value...
## 
## Variables Entered/Removed: 
## 
## - X4 added 
## - X1:X4 added 
## - log(X5) added 
## - log(X8) added 
## - log(X9) added 
## - log(X6) added 
## - log(X2) added 
## - log(X6) added 
## - log(X7) added 
## - log(X2) added 
## - X3:X4 added 
## - X3 added 
## - X1:X4 added 
## - X1 added 
## - X3:X4 added 
## - X1:X3 added 
## - X3 added 
## 
## 
## Final Model Output 
## ------------------
## 
##                         Model Summary                         
## -------------------------------------------------------------
## R                       0.986       RMSE               0.304 
## R-Squared               0.971       Coef. Var          4.779 
## Adj. R-Squared          0.962       MSE                0.093 
## Pred R-Squared          0.948       MAE                0.213 
## -------------------------------------------------------------
##  RMSE: Root Mean Square Error 
##  MSE: Mean Square Error 
##  MAE: Mean Absolute Error 
## 
##                                ANOVA                                 
## --------------------------------------------------------------------
##                Sum of                                               
##               Squares        DF    Mean Square       F         Sig. 
## --------------------------------------------------------------------
## Regression     69.356         7          9.908    107.039    0.0000 
## Residual        2.036        22          0.093                      
## Total          71.393        29                                     
## --------------------------------------------------------------------
## 
##                                   Parameter Estimates                                    
## ----------------------------------------------------------------------------------------
##       model      Beta    Std. Error    Std. Beta      t        Sig      lower     upper 
## ----------------------------------------------------------------------------------------
## (Intercept)    20.773         4.570                  4.546    0.000    11.295    30.250 
##          X4     0.480         0.045        2.197    10.600    0.000     0.386     0.574 
##     log(X5)    -4.256         1.077       -0.282    -3.952    0.001    -6.489    -2.023 
##     log(X8)     1.931         0.238        0.607     8.109    0.000     1.437     2.424 
##     log(X9)    -1.697         0.210       -0.621    -8.086    0.000    -2.132    -1.262 
##     log(X7)    -0.851         0.288       -0.255    -2.953    0.007    -1.449    -0.253 
##          X1     0.581         0.208        1.031     2.791    0.011     0.149     1.013 
##       X1:X3    -0.211         0.045       -2.332    -4.648    0.000    -0.305    -0.117 
## ----------------------------------------------------------------------------------------
k
## 
##                              Stepwise Selection Summary                               
## -------------------------------------------------------------------------------------
##                      Added/                   Adj.                                       
## Step    Variable    Removed     R-Square    R-Square      C(p)        AIC       RMSE     
## -------------------------------------------------------------------------------------
##    1       X4       addition       0.803       0.796    261.0600    68.4060    0.7087    
##    2     X1:X4      addition       0.876       0.867    156.2490    56.4454    0.5719    
##    3    log(X5)     addition       0.882       0.868    150.2850    57.0898    0.5697    
##    4    log(X8)     addition       0.889       0.871    141.5510    57.1598    0.5626    
##    5    log(X9)     addition       0.952       0.942     51.9580    34.0524    0.3779    
##    6    log(X6)     addition       0.964       0.955     36.1250    27.2246    0.3332    
##    7    log(X2)     addition       0.966       0.955     35.9080    27.9207    0.3334    
##    8    log(X6)     removal        0.952       0.940     53.2740    35.7576    0.3841    
##    9    log(X7)     addition       0.952       0.937     55.2680    37.7548    0.3927    
##   10    log(X2)     removal        0.952       0.939     53.9410    36.0450    0.3860    
##   11     X3:X4      addition       0.971       0.962     27.7710    22.5817    0.3050    
##   12       X3       addition       0.987       0.982      7.2280     1.3060    0.2118    
##   13     X1:X4      removal        0.980       0.973     15.7040    12.3534    0.2572    
##   14       X1       addition       0.985       0.979     10.1640     5.5697    0.2274    
##   15     X3:X4      removal        0.943       0.925     68.3730    42.9531    0.4283    
##   16     X1:X3      addition       0.984       0.978     10.9560     6.6232    0.2314    
##   17       X3       removal        0.971       0.962     27.5690    22.4362    0.3042    
## -------------------------------------------------------------------------------------
# plot(k)

k <- ols_step_both_aic(model_wf_mix2)
## Stepwise Selection Method 
## -------------------------
## 
## Candidate Terms: 
## 
## 1 . X1 
## 2 . X3 
## 3 . X4 
## 4 . log(X2) 
## 5 . log(X5) 
## 6 . log(X6) 
## 7 . log(X7) 
## 8 . log(X8) 
## 9 . log(X9) 
## 10 . X1:X3 
## 11 . X1:X4 
## 12 . X3:X4 
## 
## 
## Variables Entered/Removed: 
## 
## - X4 added 
## - X1:X4 added 
## - X3 added 
## - X3:X4 added 
## - log(X5) added 
## - X1:X4 removed 
## - log(X8) added 
## - log(X9) added 
## - X1:X3 added 
## - log(X5) removed 
## - log(X6) added 
## - X1 added 
## 
## No more variables to be added or removed.
k
## 
## 
##                               Stepwise Summary                              
## --------------------------------------------------------------------------
## Variable     Method      AIC       RSS      Sum Sq     R-Sq      Adj. R-Sq 
## --------------------------------------------------------------------------
## X4          addition    68.406    14.063    57.330    0.80302      0.79599 
## X1:X4       addition    56.445     8.830    62.562    0.87632      0.86715 
## X3          addition    56.315     8.225    63.168    0.88479      0.87150 
## X3:X4       addition    46.745     5.593    65.800    0.92166      0.90913 
## log(X5)     addition    44.136     4.796    66.596    0.93282      0.91882 
## X1:X4       removal     42.233     4.812    66.581    0.93260      0.92182 
## log(X8)     addition    42.149     4.489    66.904    0.93712      0.92402 
## log(X9)     addition    10.356     1.455    69.937    0.97962      0.97430 
## X1:X3       addition     4.026     1.102    70.290    0.98456      0.97964 
## log(X5)     removal      3.365     1.153    70.240    0.98385      0.97964 
## log(X6)     addition     1.186     1.003    70.390    0.98595      0.98148 
## X1          addition     0.032     0.903    70.490    0.98735      0.98254 
## --------------------------------------------------------------------------
# plot(k)

# Stepwise Regression based on p values for X4 eliminated model#
# k <- ols_step_both_p(model_wf_rm4_log)
# k
# plot(k)

# Stepwise AIC Regression for X4 eliminated model#
# k<- ols_step_both_aic(model_wf_rm4_log)
# k
# plot(k)

# Stepwise Regression based on p values for X1 eliminated model#
# k <- ols_step_both_p(model_wf_rm1_log)
# k
# plot(k)

# Stepwise AIC Regression for X1 eliminated model#
# k<- ols_step_both_aic(model_wf_rm1_log)
# k
# plot(k)
# All Possible Regression for full log model #
k <- ols_step_all_possible(model_wf_full_log)
# plot(k)
head(arrange(k, desc(adjr)))

# All Possible Regression for all log model #
k <- ols_step_all_possible(model_wf_all_log)
# plot(k)
 head(arrange(k, desc(adjr)))

# All Possible Regression for 3g log model #
#!!!!!!!!!!!! k <- ols_step_all_possible(model_wf_3g_log_inter)
# plot(k)
# head(arrange(k, desc(adjr)))
 
# All Possible Regression for mixed log model #
# k <- ols_step_all_possible(model_wf_mix2 )
# plot(k)
#  head(arrange(k, desc(adjr)))

# All Possible Regression for X4 eliminated model #
# k <- ols_step_all_possible(model_wf_rm4_log)
# k
# plot(k)

# All Possible Regression for X1 eliminated model #
# k <- ols_step_all_possible(model_wf_rm1_log)
# k
# plot(k)
#Lack of Fit F Test

ols_pure_error_anova(lm(y~X1, data = table_wf))
ols_pure_error_anova(lm(y~X4, data = table_wf))

alias(lm(y ~ as.factor(X3) + as.factor(X4) + as.factor(X5) + as.factor(X6) + as.factor(X7), data=table_wf))

alias(lm(y ~ as.factor(X1) + as.factor(X8) , data=table_wf))

alias(lm(y ~ as.factor(X4) + as.factor(X9) , data=table_wf))

alias(lm(y ~ as.factor(X3) + as.factor(X6) + as.factor(X7) + as.factor(X8) + as.factor(X9) , data=table_wf))

Final models

  • Check model aic_all_log
ols_regress(model_wf_aic_all_log )
##                         Model Summary                         
## -------------------------------------------------------------
## R                       0.993       RMSE               0.208 
## R-Squared               0.986       Coef. Var          3.273 
## Adj. R-Squared          0.982       MSE                0.043 
## Pred R-Squared          0.975       MAE                0.136 
## -------------------------------------------------------------
##  RMSE: Root Mean Square Error 
##  MSE: Mean Square Error 
##  MAE: Mean Absolute Error 
## 
##                                ANOVA                                 
## --------------------------------------------------------------------
##                Sum of                                               
##               Squares        DF    Mean Square       F         Sig. 
## --------------------------------------------------------------------
## Regression     70.394         6         11.732    270.106    0.0000 
## Residual        0.999        23          0.043                      
## Total          71.393        29                                     
## --------------------------------------------------------------------
## 
##                                   Parameter Estimates                                    
## ----------------------------------------------------------------------------------------
##       model      Beta    Std. Error    Std. Beta      t        Sig      lower     upper 
## ----------------------------------------------------------------------------------------
## (Intercept)     0.571         3.360                  0.170    0.866    -6.379     7.522 
##     log(X1)     0.726         0.036        0.967    20.107    0.000     0.651     0.800 
##     log(X3)     0.419         0.096        0.139     4.359    0.000     0.220     0.617 
##     log(X5)     1.259         0.796        0.083     1.582    0.127    -0.387     2.905 
##     log(X6)    -0.267         0.090       -0.087    -2.960    0.007    -0.454    -0.080 
##     log(X8)     1.623         0.175        0.510     9.267    0.000     1.260     1.985 
##     log(X9)    -1.375         0.154       -0.503    -8.919    0.000    -1.694    -1.056 
## ----------------------------------------------------------------------------------------
summary(model_wf_aic_all_log)
## 
## Call:
## lm(formula = log(y) ~ log(X1) + log(X3) + log(X5) + log(X6) + 
##     log(X8) + log(X9), data = table_wf)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.67722 -0.08003  0.01102  0.13879  0.25715 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.57120    3.36000   0.170  0.86650    
## log(X1)      0.72550    0.03608  20.107 4.31e-16 ***
## log(X3)      0.41866    0.09605   4.359  0.00023 ***
## log(X5)      1.25873    0.79566   1.582  0.12731    
## log(X6)     -0.26702    0.09022  -2.960  0.00702 ** 
## log(X8)      1.62253    0.17508   9.267 3.15e-09 ***
## log(X9)     -1.37489    0.15416  -8.919 6.33e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2084 on 23 degrees of freedom
## Multiple R-squared:  0.986,  Adjusted R-squared:  0.9824 
## F-statistic: 270.1 on 6 and 23 DF,  p-value: < 2.2e-16
Anova(model_wf_aic_all_log)
Sum Sq Df F value Pr(>F)
17.6   1 404    4.31e-16
0.825 1 19    0.00023 
0.109 1 2.5  0.127   
0.38  1 8.76 0.00702 
3.73  1 85.9  3.15e-09
3.45  1 79.5  6.33e-09
0.999 23            
# Collinearity Diagnostics #
ols_vif_tol(model_wf_aic_all_log)
Variables Tolerance VIF
log(X1) 0.263 3.8 
log(X3) 0.603 1.66
log(X5) 0.22  4.55
log(X6) 0.71  1.41
log(X8) 0.201 4.99
log(X9) 0.191 5.22
#Model Fit Assessment
ols_plot_diagnostics(model_wf_aic_all_log)

# Part & Partial Correlations
ols_test_correlation(model_wf_aic_all_log) # Correlation between observed residuals and expected residuals under normality.
## [1] 0.9278999
# Residual Normality Test
ols_test_normality(model_wf_aic_all_log) # Test for detecting violation of normality assumption. #If p-value is bigger, then no problem of non-normality #
## -----------------------------------------------
##        Test             Statistic       pvalue  
## -----------------------------------------------
## Shapiro-Wilk              0.8746         0.0021 
## Kolmogorov-Smirnov        0.0964         0.9180 
## Cramer-von Mises          7.0221         0.0000 
## Anderson-Darling          0.7277         0.0516 
## -----------------------------------------------
# Variable Contributions
ols_plot_added_variable(model_wf_aic_all_log)

# Residual Plus Component Plot
ols_plot_comp_plus_resid(model_wf_aic_all_log)

  • Check model aic_log_inter
summary(model_wf_aic_log_inter)
## 
## Call:
## lm(formula = log(y) ~ X1 + X2 + X3 + X4 + X5 + X7 + X8 + X9 + 
##     X1:X3 + X1:X9 + X2:X9 + X3:X8 + X4:X8 + X7:X9, data = table_wf)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.22469 -0.06872  0.00151  0.04884  0.42819 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -0.98138    1.51954  -0.646 0.528142    
## X1            2.71181    0.32987   8.221 6.15e-07 ***
## X2          -24.45201    5.63070  -4.343 0.000580 ***
## X3            0.49272    0.06581   7.487 1.93e-06 ***
## X4            0.16780    0.05415   3.099 0.007335 ** 
## X5            0.03903    0.01606   2.430 0.028137 *  
## X7            2.51458    1.25764   1.999 0.064011 .  
## X8            0.58605    0.08651   6.775 6.26e-06 ***
## X9           -0.24828    0.10461  -2.373 0.031416 *  
## X1:X3        -0.41634    0.04505  -9.242 1.40e-07 ***
## X1:X9        -0.11820    0.03621  -3.264 0.005231 ** 
## X2:X9         4.64270    1.44938   3.203 0.005925 ** 
## X3:X8        -0.05085    0.01367  -3.721 0.002050 ** 
## X4:X8         0.07075    0.01481   4.777 0.000244 ***
## X7:X9        -0.99460    0.34359  -2.895 0.011113 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.1711 on 15 degrees of freedom
## Multiple R-squared:  0.9939, Adjusted R-squared:  0.9881 
## F-statistic: 173.2 on 14 and 15 DF,  p-value: 5.77e-14
Anova(model_wf_aic_log_inter)
Sum Sq Df F value Pr(>F)
0.586   1 20     0.000446
0.277   1 9.48  0.00764 
0.00823 1 0.281 0.604   
2.99    1 102     4.36e-08
0.173   1 5.9   0.0281  
0.00974 1 0.333 0.573   
1.88    1 64.3   8.37e-07
2.59    1 88.4   1.12e-07
2.5     1 85.4   1.4e-07 
0.312   1 10.7   0.00523 
0.3     1 10.3   0.00592 
0.405   1 13.8   0.00205 
0.668   1 22.8   0.000244
0.245   1 8.38  0.0111  
0.439   15             
# Collinearity Diagnostics #
ols_vif_tol(model_wf_aic_log_inter)
Variables Tolerance VIF
X1 0.0012  835  
X2 0.032   31.3
X3 0.0141  70.8
X4 0.00668 150  
X5 0.0824  12.1
X7 0.0262  38.1
X8 0.0842  11.9
X9 0.0371  27  
X1:X3 0.00165 605  
X1:X9 0.00557 179  
X2:X9 0.0287  34.9
X3:X8 0.024   41.7
X4:X8 0.00579 173  
X7:X9 0.0111  90.3
#Model Fit Assessment
ols_plot_diagnostics(model_wf_aic_log_inter)

# Part & Partial Correlations
ols_test_correlation(model_wf_aic_log_inter) # Correlation between observed residuals and expected residuals under normality.
## [1] 0.9380444
# Residual Normality Test
ols_test_normality(model_wf_aic_log_inter) # Test for detecting violation of normality assumption. #If p-value is bigger, then no problem of non-normality #
## -----------------------------------------------
##        Test             Statistic       pvalue  
## -----------------------------------------------
## Shapiro-Wilk              0.898          0.0075 
## Kolmogorov-Smirnov        0.1576         0.4037 
## Cramer-von Mises          8.0824         0.0000 
## Anderson-Darling          0.8459         0.0259 
## -----------------------------------------------
# Variable Contributions
ols_plot_added_variable(model_wf_aic_log_inter)

# Residual Plus Component Plot
ols_plot_comp_plus_resid(model_wf_aic_log_inter)

  • Check model mixed
summary(model_wf_aic_mix2 )
## 
## Call:
## lm(formula = log(y) ~ X1 + X3 + X4 + log(X7) + log(X8) + log(X9) + 
##     X1:X3 + X1:X4, data = table_wf)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.56967 -0.05648 -0.01439  0.08018  0.33666 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  2.04446    0.34314   5.958 6.50e-06 ***
## X1           1.45925    0.23161   6.301 3.01e-06 ***
## X3           0.27464    0.03851   7.131 4.94e-07 ***
## X4           0.42303    0.04203  10.066 1.73e-09 ***
## log(X7)     -0.42799    0.20817  -2.056   0.0524 .  
## log(X8)      1.62847    0.16269  10.010 1.90e-09 ***
## log(X9)     -1.40518    0.14700  -9.559 4.24e-09 ***
## X1:X3       -0.39830    0.05278  -7.546 2.07e-07 ***
## X1:X4        0.02624    0.01181   2.222   0.0374 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2116 on 21 degrees of freedom
## Multiple R-squared:  0.9868, Adjusted R-squared:  0.9818 
## F-statistic: 196.7 on 8 and 21 DF,  p-value: < 2.2e-16
Anova(model_wf_aic_mix2 )
Sum Sq Df F value Pr(>F)
0.372 1 8.32 0.00888 
0.167 1 3.72 0.0674  
5.54  1 124    2.93e-10
0.189 1 4.23 0.0524  
4.49  1 100    1.9e-09 
4.09  1 91.4  4.24e-09
2.55  1 56.9  2.07e-07
0.221 1 4.94 0.0374  
0.94  21            
# Collinearity Diagnostics #
ols_vif_tol(model_wf_aic_mix2)
Variables Tolerance VIF
X1 0.00371 269   
X3 0.0631  15.9 
X4 0.017   59   
log(X7) 0.161   6.2 
log(X8) 0.239   4.18
log(X9) 0.217   4.61
X1:X3 0.00184 543   
X1:X4 0.00401 249   
#Model Fit Assessment
ols_plot_diagnostics(model_wf_aic_mix2)

# Part & Partial Correlations
ols_test_correlation(model_wf_aic_mix2) # Correlation between observed residuals and expected residuals under normality.
## [1] 0.9628432
# Residual Normality Test
ols_test_normality(model_wf_aic_mix2) # Test for detecting violation of normality assumption. #If p-value is bigger, then no problem of non-normality #
## -----------------------------------------------
##        Test             Statistic       pvalue  
## -----------------------------------------------
## Shapiro-Wilk              0.9393         0.0868 
## Kolmogorov-Smirnov        0.1411         0.5422 
## Cramer-von Mises          7.0915         0.0000 
## Anderson-Darling          0.5183         0.1733 
## -----------------------------------------------
# Variable Contributions
ols_plot_added_variable(model_wf_aic_mix2)

# Residual Plus Component Plot
ols_plot_comp_plus_resid(model_wf_aic_mix2)

  • Check model all log interaction
summary(model_wf_aic_all_log_inter)
## 
## Call:
## lm(formula = log(y) ~ log(X1) + log(X2) + log(X3) + log(X4) + 
##     log(X5) + log(X6) + log(X7) + log(X8) + log(X9) + log(X1):log(X3) + 
##     log(X1):log(X8) + log(X1):log(X9) + log(X2):log(X8) + log(X2):log(X9) + 
##     log(X3):log(X9) + log(X4):log(X8) + log(X4):log(X9) + log(X5):log(X8) + 
##     log(X5):log(X9) + log(X6):log(X8) + log(X6):log(X9) + log(X7):log(X9), 
##     data = table_wf)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.114032 -0.038669 -0.003953  0.026220  0.160039 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)   
## (Intercept)     -16.0272    13.9469  -1.149  0.28823   
## log(X1)           1.1577     0.6966   1.662  0.14045   
## log(X2)          -0.7328     0.3383  -2.166  0.06698 . 
## log(X3)           1.6656     1.7630   0.945  0.37625   
## log(X4)          -2.3936     2.9633  -0.808  0.44581   
## log(X5)           5.0279     3.4728   1.448  0.19094   
## log(X6)          -0.1583     0.3004  -0.527  0.61463   
## log(X7)          -0.4075     0.5183  -0.786  0.45748   
## log(X8)          44.9189    24.1746   1.858  0.10550   
## log(X9)         -37.0656    19.1702  -1.934  0.09443 . 
## log(X1):log(X3)   0.8234     0.7067   1.165  0.28218   
## log(X1):log(X8)   1.2421     0.6407   1.939  0.09372 . 
## log(X1):log(X9)  -0.9995     0.6739  -1.483  0.18156   
## log(X2):log(X8)   1.0890     0.3972   2.742  0.02885 * 
## log(X2):log(X9)  -0.7095     0.3415  -2.078  0.07633 . 
## log(X3):log(X9)   0.4112     0.3562   1.154  0.28626   
## log(X4):log(X8)  -3.0288     0.8494  -3.566  0.00915 **
## log(X4):log(X9)   2.1744     0.9286   2.342  0.05172 . 
## log(X5):log(X8)  -7.9687     5.5620  -1.433  0.19504   
## log(X5):log(X9)   6.7951     4.4498   1.527  0.17059   
## log(X6):log(X8)   0.4027     0.2891   1.393  0.20622   
## log(X6):log(X9)  -0.5057     0.2651  -1.908  0.09807 . 
## log(X7):log(X9)   0.4637     0.3759   1.234  0.25719   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.1252 on 7 degrees of freedom
## Multiple R-squared:  0.9985, Adjusted R-squared:  0.9936 
## F-statistic: 206.6 on 22 and 7 DF,  p-value: 7.713e-08
Anova(model_wf_aic_all_log_inter)
Sum Sq Df F value Pr(>F)
0.281    1 17.9    0.00386 
0.00972  1 0.619  0.457   
0.0102   1 0.648  0.447   
0.000172 1 0.0109 0.92    
0.0311   1 1.98   0.202   
0.068    1 4.33   0.0759  
0.00234  1 0.149  0.711   
1.7      1 108      1.65e-05
1.73     1 111      1.53e-05
0.0213   1 1.36   0.282   
0.059    1 3.76   0.0937  
0.0345   1 2.2    0.182   
0.118    1 7.52   0.0288  
0.0677   1 4.32   0.0763  
0.0209   1 1.33   0.286   
0.199    1 12.7    0.00915 
0.086    1 5.48   0.0517  
0.0322   1 2.05   0.195   
0.0366   1 2.33   0.171   
0.0304   1 1.94   0.206   
0.0571   1 3.64   0.0981  
0.0239   1 1.52   0.257   
0.11     7              
# Collinearity Diagnostics #
ols_vif_tol(model_wf_aic_all_log_inter)
Variables Tolerance VIF
log(X1) 0.000255 3.92e+03
log(X2) 0.00469  213       
log(X3) 0.000646 1.55e+03
log(X4) 4.36e-05 2.29e+04
log(X5) 0.00416  240       
log(X6) 0.0231   43.2     
log(X7) 0.00911  110       
log(X8) 3.8e-06  2.63e+05
log(X9) 4.47e-06 2.24e+05
log(X1):log(X3) 0.000108 9.3e+03 
log(X1):log(X8) 0.000275 3.64e+03
log(X1):log(X9) 0.000202 4.94e+03
log(X2):log(X8) 0.000851 1.18e+03
log(X2):log(X9) 0.000809 1.24e+03
log(X3):log(X9) 0.00282  355       
log(X4):log(X8) 0.000283 3.53e+03
log(X4):log(X9) 0.00019  5.25e+03
log(X5):log(X8) 4.17e-06 2.4e+05 
log(X5):log(X9) 4.82e-06 2.07e+05
log(X6):log(X8) 0.0203   49.4     
log(X6):log(X9) 0.0191   52.3     
log(X7):log(X9) 0.00632  158       
#Model Fit Assessment
ols_plot_diagnostics(model_wf_aic_all_log_inter)

# Part & Partial Correlations
ols_test_correlation(model_wf_aic_all_log_inter) # Correlation between observed residuals and expected residuals under normality.
## [1] 0.9830352
# Residual Normality Test
ols_test_normality(model_wf_aic_all_log_inter) # Test for detecting violation of normality assumption. #If p-value is bigger, then no problem of non-normality #
## -----------------------------------------------
##        Test             Statistic       pvalue  
## -----------------------------------------------
## Shapiro-Wilk              0.9695         0.5254 
## Kolmogorov-Smirnov        0.1176         0.7579 
## Cramer-von Mises          8.8358         0.0000 
## Anderson-Darling          0.3654         0.4134 
## -----------------------------------------------
# Variable Contributions
ols_plot_added_variable(model_wf_aic_all_log_inter)

# Residual Plus Component Plot
ols_plot_comp_plus_resid(model_wf_aic_all_log_inter)